Universal fast-flux control of low-frequency qubits

ABSTRACT

Methods for initializing a qubit into a pure state, reading the qubit, and arbitrarily rotating the qubit into any quantum state complete in times shorter than the qubit&#39;s typical dephasing and relaxation times. These methods provide universal single-qubit control and may be used to implement quantum gates with high fidelity. The methods may be implemented with superconducting qubits, such as heavy fluxonium, and do not rely on a three-dimensional cavity for suppressing spontaneous emission. Therefore, the methods may be implemented using smaller two-dimensional architectures commonly used for superconducting circuits. The methods also work with low-frequency qubits, i.e., qubits for which the energy spacing between the two quantum-computational states is less than the mean thermal energy of a surrounding bath. This reduces the cooling requirements of the qubit while maintaining fidelity.

RELATED APPLICATIONS

This application is a 35 U.S.C. § 371 filing of InternationalApplication No. PCT/US2021/018428, filed on Feb. 17, 2021, which claimspriority to U.S. Provisional Patent Application No. 62/978,017, filed onFeb. 18, 2020. Each of these applications is incorporated herein byreference in its entirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under grant numberW911NF1910016 awarded by the Army Research Office. The government hascertain rights in the invention.

BACKGROUND

Superconducting circuits are among the most promising of qubittechnologies for scalable quantum computation due to their longcoherence times, high gate fidelities, and processor size (i.e., thenumber of qubits that can be manipulated simultaneously).

SUMMARY

Many prior-art superconducting quantum processors implement qubits astransmons, whose coherence times have improved by nearly an order ofmagnitude due to new methods that decrease the effects of environmentalnoise. One of the simplest of superconducting circuits, a transmon canbe modeled as a weakly anharmonic oscillator whose energy states arejoined via large transition dipole matrix elements. Transmons trade offdecreased sensitivity to charge-induced-noise dephasing with increasedsensitivity to decay.

Flux qubits, and in particular fluxonium, are promising types ofsuperconducting circuits that offer many advantages over transmons. Someof the benefits of fluxonium include a rich energy-level structure,natural protection from charge-noise-induced relaxation and dephasing,and compatibility with circuit quantum electrodynamics (QED).

Fluxonium is created by shunting a flux qubit with a large inductance.Further shunting of the qubit with a large capacitance results in “heavyfluxonium”, which advantageously increases relaxation times. However, achallenge with using flux qubits in large-scale superconductingprocessors is that standard microwave control techniques result in slowgates that cannot complete before dephasing and relaxation take effect.

The present embodiments include fast methods for controllingsuperconducting qubits, and therefore may be used with superconductingquantum computers (e.g., those based on fluxonium qubits). Here, “fast”means that the methods complete in times much shorter than typicaldephasing and relaxation times, and thus high fidelities (e.g., 99%) canbe reliably achieved. Advantageously, these methods do not rely on athree-dimensional (3D) cavity for suppressing spontaneous emission, andtherefore can be implemented using smaller two-dimensional (2D)architectures commonly used for superconducting circuits. The presentembodiments include methods for initializing a superconducting qubitinto a pure state (e.g., at the beginning of a quantum gate) and methodsfor reading-out the quantum state of the superconducting qubit (e.g., atthe end of the quantum gate). Finally, the present embodiments alsoinclude methods based on fast pulses for arbitrarily rotating a singlesuperconducting qubit into any quantum state, thereby providinguniversal single-qubit control.

Advantageously, the present embodiments can work when the energy spacingbetween the two quantum-computational states of the qubit is less thanthe equivalent mean thermal energy of a surrounding bath. Prior-artsuperconducting qubits are challenged by small energy spacings becausethe resulting thermal occupation of both quantum-computational statesreduces fidelity. By overcoming the limitations of this thermaloccupation, the present embodiments relieve the requirements forcryogenic cooling (i.e., the qubit does not need to be cooled totemperatures as low as those used in the prior art, while maintainingfidelity).

For clarity in the following discussion, examples are described withrespect to fluxonium qubits. However, it should be appreciated that thepresent embodiments may be used with any kind of flux qubit, i.e., a“heavy” flux qubit, a “light” flux qubit, or any other type of qubitwhose properties are controllable with an applied magnetic flux.Furthermore, the present embodiments may also be used with another typeof qubit without departing from the scope hereof. For example, thepresent embodiments can be implemented with a charge qubit (e.g., acooper pair box). The present embodiments can also be implemented with aqubit containing a voltage-controllable Josephson junction (e.g., agatemon qubit, or flux-like qubits where the Josephson junction isvoltage-controlled).

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 is a schematic diagram of a quantum system in which a fluxoniumqubit is capacitively coupled to a readout resonator, in an embodiment.

FIG. 2 is a plot showing potential energy and wavefunctions of thefluxonium qubit of FIG. 1 as a function of junction phase.

FIG. 3 is an energy-level diagram of the quantum system of FIG. 1 , inan embodiment.

FIG. 4 is a plot of magnetic flux as a function of time, illustrating afast magnetic pulse for single-qubit rotation of the fluxonium qubit ofFIG. 1 , in an embodiment.

FIG. 5 is a plot of magnetic flux as a function of time, illustrating amagnetic-pulse sequence for universal single-qubit rotation of thefluxonium qubit of FIG. 1 , in an embodiment.

FIG. 6 shows one example of a complex pulse that crosses the nominalflux several times, in an embodiment.

FIG. 7A shows an optical microscope image of a fluxonium circuit coupledto a readout resonator along with flux and input-output lines (leftpanel), a scanning electron micrograph of the large junction array andthe small Josephson junction (center panel), and a zoom-in view of thesmall junction (right panel), in an embodiment.

FIG. 7B shows a circuit diagram for the heavy-fluxonium qubit, in anembodiment.

FIG. 7C shows an energy-level diagram of the heavy-fluxonium at theflux-frustration point. FIG. 7C also shows the potential well, the firstsix energy eigenstates, and dashed lines representing the wavefunctionsfor the first four levels.

FIG. 8A shows a level diagram for reset and readout protocols, inembodiments. Reset is performed by simultaneously driving both |g0

→|h0

and |h0

→|e1

transitions, as indicated by the double-headed arrows. The spontaneousphoton decay along |e1

→|e0

provides a directional transition that completes the reset. An |e0

→|f0

π pulse is applied before the readout to boost the output signal.

FIG. 8B shows Rabi oscillations between |e

and |f

for different initial state preparations. Squares indicate data pointsfor which the initial state was prepared in |e

before the |e

↔|f

Rabi flopping. Circles indicate data points for which the initial statewas in the thermal equilibrium state. Diamonds indicate data points forwhich the initial state was |e

.

FIG. 9A is a plot of energy relaxation time (T₁) as a function of flux,along with the theoretical limits set by dielectric (T₁ ^(Cap)),inductive (T₁ ^(Ind)), Purcell (T₁ ^(Purcell)), and the combined loss(T₁ ^(Total)). The inset shows the decay of P(|e

) to 0.495 after preparing the qubit in |g

, |e

at the flux-frustration point.

FIG. 9B is a plot of echo decay time T_(2e) as a function of flux nearthe flux-frustration point. The inset shows an echo measurement at theflux-frustration point.

FIG. 10A shows a generic pulse scheme, in an embodiment. We use net-zeroflux pulses for our native gates (Y/2 and Y). They are constructed usingthree sections, a positive triangular pulse with amplitude A and widthΔt_(p) on the fast-flux line, an idling period of Δt_(z) and finallyanother triangular pulse identical to the first one but with a negativeamplitude.

FIG. 10B shows energy levels of the computational space as a function ofexternal flux (Φ_(ext)) showing how the fast-flux pulse changes theenergies of the instantaneous eigenstates.

FIG. 10C shows expectation values of σ_(x), σ_(y), and σ_(z) as afunction of pulse parameters Δt_(z) and A. These 2D sweeps are used todetermine the optimal parameter for the Y/2 and Y gates. In FIG. 10C,the star ★ indicates the parameters for a Y/2 gate.

FIG. 10D shows trajectories of three distinct initial states |0

, (|0

+|1

)/√{square root over (2)} and (|0

+i|1

)/√{square root over (2)} on the Bloch sphere when a Y/2 gate isapplied.

FIG. 10E is a plot comparing standard randomized benchmarking (RB,circles) and interleaved RB for Z/2 (triangles), Y/2 (diamonds) and X/2(squares) gates. FIG. 10E was generated from 75 randomized gatesequences averaged over 10,000 times. The average gate fidelity is

_(avg)=0.9980 and the individual gate fidelities are

_(Z/2)=0.9999,

_(Y/2)=0.9992 and

_(X/2)=0.9976. The uncertainties in all fidelities are smaller than theleast significant digit.

FIG. 11 is a wiring diagram inside the dilution refrigerator. Outsidethe dilution fridge, there are ˜16 dB of attenuation and a DC block onthe RF flux line, and an ultra-low pass (˜1 Hz) RC filter on the DC fluxline. The total attenuation on the RF flux line proved important forboth the T₁ and T₂ of the qubit, likely due to reduction in noise fromthe arbitrary waveform generator (Agilent 81180A).

FIG. 12 shows simulated expectation values of σ_(x), σ_(y) and σ_(z) asa function of pulse parameters Δt_(z) and A with Δt_(p)=4.76 ns. Thesimulation shows good agreement with the experimental data shown in FIG.4C.

FIG. 13 shows the population in the |e

state as a function of the length of the reset pulse. The population wasmeasured after simultaneously driving |g0

→|h0

and |h0

→|e1

transitions for different lengths of time. Reset of the state wasachieved in ˜5 μs.

FIG. 14A is histogram of the lowest four fluxonium states |g

, |e

, |f

, and |h

. The |g

-|f

readout fidelity is ˜50%.

FIG. 14B shows the distribution of all single shot data from the lowestfour fluxonium states on the IQ plane.

DETAILED DESCRIPTION

FIG. 1 is a schematic diagram of a quantum system 100 in which afluxonium qubit 120 is coupled, via a coupling capacitor 106, to areadout resonator 110. The fluxonium qubit 120 is an example of heavyfluxonium in which a small-area Josephson junction 122 with inductanceL_(J) is shunted by a capacitor 124 with capacitance C_(q). Thefluxonium qubit 120 also includes a superinductor 126 formed from anarray of large-area Josephson junctions that shunts the Josephsonjunction 122 with an inductance L_(JA). The superinductor 126 has atotal Josephson energy E_(JA) that is much larger than a total chargingenergy E_(CA) of the superinductor 126 (i.e., E_(JA)>>E_(CA)) to ensurethat charge dispersion for each junction of the superinductor 126 issmall, and thus that the superinductor 126 serves only as an inductor.

A Hamiltonian H_(f) for the fluxonium qubit 120 is

$\begin{matrix}{{H_{f} = {{{- 4}E_{C}\frac{d^{2}}{d\varphi^{2}}} - {E_{J}{\cos\left( {\varphi - {2\pi\frac{\Phi_{ext}}{\Phi_{0}}}} \right)}} + {\frac{1}{2}E_{L}\varphi^{2}}}},} & (1)\end{matrix}$

where Φ is the phase across the Josephson junction 122,E_(C)=e²/(2C_(q)) is the charging energy of the capacitor 124 (e beingthe charge of the electron), E_(J)=Φ₀ ²/(2L_(J)) is the Josephson energyof the Josephson junction 122, E_(L)=Φ₀ ²/(2L_(JA)) is the inductiveenergy of the superinductor 126, Φ_(ext) is external magnetic flux 130threading a loop 132 formed by the superinductor 126 and the Josephsonjunction 122, and Φ=h/(2e) is the superconducting magnetic flux quantum(h being Planck's constant).

The readout resonator 110 is represented in FIG. 1 as a readout inductor112 that resonates with first and second readout capacitors 114 and 116.An inductance L_(R) of the readout inductor 112, and a parallelcapacitance C_(R) of the readout capacitors 114, 116, are selected suchthat the readout resonator 110 has a resonant frequency that is highcompared to a temperature of an environment in which the quantum system100 is located and operates. For example, the quantum system 100 may bemounted inside of a cryostat that cools the quantum system 100 to atemperature less than 50 mK. At 10 mK, the peak of the blackbodyspectrum occurs near 600 MHz (as estimated using Wien's displacementlaw). Accordingly, the resonant frequency of the readout resonator 110should be made higher than this value (e.g., 5 GHz, or more) to minimizeexcitation of the readout resonator 110 by thermal background photons.At temperatures less than 10 mK, the resonant frequency of the readoutresonator 110 may be reduced accordingly.

To measure a state of the fluxonium qubit 120, the readout resonator 110may be probed (e.g., via two-tone spectroscopy) using a first microwavetransmission line 140 coupled to the readout resonator 110 via an inputcapacitor 104, and with a second microwave transmission line 142 coupledto the readout resonator 110 via an output capacitor 102. Each of themicrowave transmission lines 140 and 142 may be a coaxial transmissionline, or a planar transmission line (e.g., microstrip) co-fabricatedwith the quantum system 100 on a common substrate. The quantum system100 may be alternatively or additionally coupled to one or more othersuperconducting quantum components (e.g., one or more additionalresonators 110, one or more additional fluxonium qubits 120, one or moreother additional superconducting qubits of another type, etc.).

FIG. 2 is a plot showing potential energy 202 and wavefunctions 210,212, 214, and 216 of the fluxonium qubit 120 as a function of thejunction phase φ, when the loop 132 is threaded with externallygenerated magnetic flux 130 equal to one-half of the superconductingmagnetic flux quantum (i.e., φ_(ext)=φ₀/2, also referred to as the“flux-frustration point”). The potential energy 202, corresponding tothe second and third terms on the right-hand side of Eqn. 1, exhibit apair of potential-energy wells near ±π. The fluxonium qubit 120 has aground state |g

with a ground-state wavefunction 210 and a ground-state energy E_(g), afirst excited state |e

with a first excited-state wavefunction 212 and a first excited-stateenergy E_(e), a second excited state |f

with a second excited-state wavefunction 214 and a second excited-stateenergy E_(f), and a third excited state |h

with a third excited-state wavefunction 216 and a third excited-stateenergy E_(h). Each of the states |g

, |e

, |f

, and |h

is an energy eigenstate of the Hamiltonian H_(f) of Eqn. 1.

Quantum computation with the fluxonium qubit 120 may be implemented withthe ground state |g

and the first excited state |e

. These states are also referred to herein as the quantum-computationalstates |g

and |e

. Tunneling through a center peak 218 of the potential energy 202removes a degeneracy between these two states, resulting in an energyspacing Δ₁=E_(e)−E_(g). In the example of FIG. 2 , Δ₁ is only 14 MHz,much less than the frequency of the environmental temperature. Toachieve such a small value for Δ₁ (i.e., weak tunneling through thecenter peak 218), the height of the center peak 218 can be made large byselecting the ratio E_(J)/E_(C) to be greater than one. In the exampleof FIG. 2 , E_(J)/E_(C)≈7. However, the fluxonium qubit 120 can beconfigured with a ratio E_(J)/E_(C) that is even larger, wherein Δ₁decreases accordingly. Alternatively, the fluxonium qubit 120 can beconfigured with a smaller ratio E_(J)/E_(C) without departing from thescope hereof. Alternatively, the fluxonium qubit 120 can be configuredwith an additional Josephson junction that allows E_(J)/E_(C) to beelectronically controlled.

Associated with small values of Δ₁ is a suppression of transitionsbetween the states |g

and |e

. That is, the transition dipole between the states |g

and |e

is nearly zero. Accordingly, the first excited state |e

is metastable, which can be seen in FIG. 1 by the fact that the overlapof the wavefunctions 210 and 212, when integrated over all values of φ,is small. Transitions between |e

and |g

are fluxon-like inter-well transitions. Accordingly, the first excitedstate |e

is also referred to herein as the metastable fluxon state |e

.

Each of the two potential-energy wells in FIG. 2 is deep enough tosupport an additional eigenstate, shown in FIG. 2 as the second andthird excited states |f

and |h

. Transitions between the ground state |g

and the third excited state |h

are intra-well transitions that are plasmon-like. Transitions betweenthe first excited state |e

and the second excited state |f

are also plasmon-like. Unlike the suppressed fluxon transitions betweenthe states |g

and |e

, each of these plasmon-like transitions is strong, and thus can bereadily driven via applied microwave fields. As shown in FIG. 2 , anenergy spacing Δ₂=E_(f)−E_(e) between the second excited state |f

and the first excited state |e

is larger than Δ₁. Similarly, an energy spacing Δ₃×E_(h)−E_(f) betweenthe third excited state |h

and the second excited state |f

is greater than Δ₁ due to the increasing tunneling between these statesnear the top of the center peak 218. Nevertheless, Δ₃ is still less thanΔ₂. In the example of FIG. 2 , Δ₂ is 2974 MHz and Δ₃ is 194 MHz.Accordingly, the second excited state |f

is also referred to herein as the first plasmon excited state |f

, and the third excited state |h

is also referred to herein as the second plasmon excited state |h

.

FIG. 3 is an energy-level diagram of the quantum system 100 of FIG. 1 .In FIG. 3 , each energy eigenstate of the quantum system 100 is acomposite (i.e., tensor product) of one of the energy eigenstates of thefluxonium qubit 120 (i.e., one of the states |g

, |e

, |f

, and |h

) and a quantum state of the resonator 110 characterized by a number ofphotons in the resonator 110. Accordingly, FIG. 3 shows a compositestate |g0

in which the fluxonium qubit 120 is in the ground state |g

and there are no photons in the resonator 110. Similarly, FIG. 3 shows acomposite state |e0

in which the fluxonium qubit 120 is in the first excited state |e

and there are no photons in the resonator 110. FIG. 3 also showscomposite states |f0

and |h0

, which may be interpreted similarly.

FIG. 3 also shows a composite state |e1

in which the fluxonium qubit 120 is in the first excited state |e

and there is one photon in the resonator 110. The resonator 110 has arelatively low Q (e.g., less than 1000) and thus, when the quantumsystem 100 is in the state |e1

, the resonator 110 will rapidly decay, emitting a spontaneous photon306 that leaves the quantum system 100 in the state |e0

. Note that the quantum system 100, when in the state |e1

, is highly unlikely to decay to the state |g0

since such a decay involves a fluxon-like transition for the fluxoniumqubit 120. As described above, such fluxon-like transitions are highlysuppressed.

While FIG. 1 shows the fluxonium qubit 120 coupled to the readoutresonator 110, the fluxonium qubit 120 may be alternatively coupled toanother type of energy-dissipating device that couples the energy of thespontaneous photon 306 into a surrounding cold bath to reduce theentropy of the fluxonium qubit 120. Other examples of such energydissipaters include 3D microwave cavities and microwave transmissionlines.

FIG. 3 also illustrates a reset method for initializing the quantumsystem 100 into a pure state. Since the quantum system 100 resides in anenvironment whose temperature is large compared to Δ₁, yet smaller thanΔ₂, the fluxonium qubit 120, when thermally equilibrated withenvironment, will be in a nearly evenly mixed state (i.e., a nearly evenstatistical ensemble of the pure states |g

and |e

). Advantageously, the reset method described here transfers thefluxonium qubit 120 from the mixed state into a pure state (i.e., eitherone of the states |g

and |e

) that can be subsequently controlled to prepare the fluxonium qubit120, with high fidelity, in whatever state is needed (as based on thequantum algorithm or gate at hand). Thus, the reset method may be usedas a first step for universal single-qubit control.

The initial mixed state of the fluxonium qubit 120 can be described asan ensemble whose density matrix operator has the form ρ=p_(q)|g

g|+p_(e)|e

e|, where p_(g) is the fraction of the ensemble in the state |g

and p_(e) is the fraction of the ensemble in the state |e

. Thus, the operator ρ has a g-term and an e-term. As shown in FIG. 3 ,the quantum system 100 may be driven with a first microwave field 302that couples the states |g0

and |h0

. For example, a frequency of the first microwave field 302 may beselected to be resonant with the transition between the states |g0

and |h0

. The first microwave field 302 excites the fraction p_(g) of theensemble in the state |g

to the state |h

, while leaving the fraction p_(e) of the ensemble in the state |e

undisturbed.

Also shown in FIG. 3 , the quantum system 100 may also be driven with asecond microwave field 304 that couples the states |h0

and |e1

. For example, a frequency of the first microwave field 302 may beselected to be resonant with the transition between the states |h0

and |e1

. The microwave fields 302 and 304 may be applied simultaneously. Thesecond microwave field 304 excites the fraction p_(g) of the ensemblepreviously transferred to the state |h0

up to the state |e1

. Once excited, the fraction rapidly decays into the |e0

state by emitting the spontaneous photon 306. After the decay, all theensemble is in the |e

state of the fluxonium qubit 120, and thus the quantum system 100 is ina pure state. A third microwave field 310 may be subsequently applied totransfer the quantum system 100 into the pure state |g0

. For example, the third microwave field 310 may be a π pulse thatcoherently transfers the quantum system 100 from the pure state |e0

into the pure state |g0

.

A first speed with which the first microwave field 302 excites thefraction p_(g) from the state |g0

to the state |h0

depends on a first Rabi frequency that is equal to the product of anamplitude of the first microwave field 302 and a transition dipolemoment between the states |g0

and |h0

. Similarly, a second speed with which the second microwave field 304excites the fraction p_(g) from the state |h0

to the state let) depends on a second Rabi frequency that is equal tothe product of an amplitude of the second microwave field 304 and atransition dipole moment between the states |h0

and |e1

. Thus, the amplitudes of the microwave field 302, 304 may be chosensuch that the fraction p_(g) is excited to the state |e1

, and subsequently decays to the state |e0

, in a time that is short compared to both a relaxation time T₁ and adephasing time T₂ of the fluxonium qubit 120 (see experimentaldemonstration below). By contrast, the reset method was experimentallydemonstrated by applying the microwave fields 302 and 304 to thefluxonium qubit 120 for 15 μs, followed by a 10 μs waiting period (i.e.,in the absence of the microwave fields 302 and 304) to allow thespontaneous photon 306 to be emitted. A π pulse was the applied, afterwhich a fidelity of 99% was obtained (i.e., 99% of the ensemble was inthe state |g0

, with the remaining 1% of the ensemble in other states). Thecorresponding temperature of the fluxonium qubit 120 was only 0.145 mK,lower than the ambient temperature by a factor of 100. Since it isexperimentally challenging to produce a perfectly pure state, the resetmethod is described herein as generating an “approximately” pure state.

It is assumed in the previous discussion that the loop 132 of thefluxonium qubit 120 is continuously threaded by the external magneticflux 130 during the entire reset method (and subsequent π pulse, ifincluded), as the external magnetic flux 130 is needed to generate theenergy-level structure shown in FIG. 3 . While the above discussiondescribes the reset method with respect to the fluxonium qubit 120,those trained in the art will recognize that the reset method can beapplied to another type of qubit, provided that the qubit has, orincludes, a similar energy-level structure to that shown in FIG. 3 .Thus, the reset method is not limited to fluxonium qubits, but may beused with other types of flux qubits, other types ofsuperconducting-circuit qubits (e.g., phase qubits), and evennon-superconducting-circuit qubits (e.g., ions, atoms, nitrogen-vacancycenters, etc.). In some of these other types of qubits, one or more ofthe microwave fields 302, 304, and 308 may need to be replaced with anoscillating electromagnetic field in a different part of theelectromagnetic spectrum (e.g., infrared, visible, ultraviolet, etc.).

FIG. 3 also illustrates a readout method for distinguishing between thequantum-computational states |_(g)

and |e

of the fluxonium qubit 120. The fluxonium qubit 120, when coupled withthe readout resonator 110, dispersively shifts the resonant frequency ofthe readout resonator 110 by an amount that depends on the state (i.e.,|g

or |e

) of the fluxonium qubit 120. Thus, the fluxonium qubit 120 induces, inthe readout resonator 110, a first dispersive frequency shift when inthe |g

state, and a second dispersive frequency shift when in the |e

state. A differential shift, equal to the different between the firstand second dispersive frequency shifts, increases as the energies of thestates |g

and |e

approach that of the excited resonator 110 (i.e., as the energies of thestates |g0

and |e0

increase toward the energy of the state |e1

).

For the quantum system 100, the energy gap between each of thelower-energy states |g0

and |e0

, and the excited state |e1

is so large that the differential shift is too small to discern betweenthe states |g0

and |e0

. To enhance the interaction with the readout resonator 110, a π pulse308 may be applied between the states |e0

and |f0

, as part of the readout method, to coherently transfer the populationof the state |e0

into the state |f0

. The goal of discerning between the states |g0

and |e0

is now implemented by discerning between the states |g0

and |f0

. Since the energy of the state |f0

is closer to that of the excited state |e1

, the second dispersive frequency shift increases. This, in turn,increases the differential shift, making it easier to discern betweenthe states |g0

and |f0

using dispersive readout via the readout resonator 110. The π pulse 308may be alternatively configured to coherently transfer the population inthe state |g0

state to the state |h0

, wherein the readout resonator 110 is used to discern between thestates |e0

and |h0

.

One advantage of the readout method described above is that quantumcomputation is performed in the states |g0

and |e0

, which benefits from the large detunings to the lowest excited statesof the resonator 110. Thus, during quantum computation, heating of thefluxonium qubit 120 due to coupling with the resonator 110 is minimized,helping to preserve the long relaxation and dephasing times.

It is assumed in the previous discussion that the loop 132 of thefluxonium qubit 120 is continuously threaded by the external magneticflux 130 during the entire readout method, as the external magnetic flux130 is used to generate the energy-level structure shown in FIG. 3 .However, the external magnetic flux 130 may be changed during thereadout method. For example, a first magnetic flux 130 may be appliedprior to and/or during the π pulse 308. Afterwards, a second magneticflux 130 (with a magnitude different than that of the first magneticflux 130) may be applied during sequent dispersive readout via thereadout resonator 110.

While the above discussion describes the readout method with respect tothe fluxonium qubit 120, those trained in the art will recognize thatthe readout method can be applied to another type of qubit, providedthat the qubit has, or includes, a similar energy-level structure tothat shown in FIG. 3 . Thus, the reset method is not limited tofluxonium qubits, but may be used with other types of flux qubits, othertypes of superconducting-circuit qubits (e.g., phase qubits), and evennon-superconducting-circuit qubits (e.g., ions, atoms, nitrogen-vacancycenters, etc.). In some of these other types of qubits, one or more ofthe microwave fields 302, 304, and 308 may need to be replaced with anoscillating electromagnetic field in a different part of theelectromagnetic spectrum (e.g., infrared, visible, ultraviolet, etc.).

FIG. 4 is a plot of magnetic flux (I) as a function of time,illustrating a fast magnetic pulse 402 for single-qubit rotation of thefluxonium qubit 120. The fluxonium qubit 120 may begin (i.e., before aninitial time t₁) in any qubit state, i.e., any linear combination of thequantum-computational states |g

and |e

. The fluxonium qubit 112 is continuously threaded with externallygenerated magnetic flux 130 at a nominal value. Starting at the initialtime t₁, and lasting until a second time t₂=t₁+Δt, the pulse 402 isapplied to the fluxonium qubit 120 by deviating the magnetic flux Φ awayfrom the nominal value. In the example of FIG. 4 , the magnetic flux Φis increased above the nominal value by a pulse amplitude A. The valueΔt is a pulse duration of the pulse 402. While FIG. 4 shows the magneticflux Φ returning to the same nominal value after the pulse 402 (i.e.,starting at time t₂), the magnetic flux Φ may return to a differentnominal value than that before the pulse 402 (i.e., prior to time t₁).

The magnetic pulse 402 is “fast” in the sense that the pulse duration Δtis much less than the Larmor period (i.e., the inverse of the Larmorfrequency ω_(q)) of the fluxonium qubit 120. The Larmor frequency ω_(q)is given by the energy splitting of the states |g

and |e

when the magnetic flux Φ is at the nominal value Φ₀/2. In theexperimental results presented below, ω_(q)=14 MHz, corresponding to aLarmor period of 71 ns. A pulse duration Δt of approximately 2 ns wassuccessfully demonstrated, more than an order of magnitude less than theLarmor period. In one embodiment, the pulse duration Δt is less thanone-fourth of the Larmor period.

Representing the qubit state as a Bloch vector on the Bloch sphere, thepulse 402 rotates the Bloch vector by an angle θ about the x axis of theBloch sphere (see FIG. 11D). Due to the finite value of the pulseduration Δt, the pulse 402 also rotates the Bloch vector about the zaxis of the Bloch sphere, albeit at a slower rate than rotation aboutthe x axis. The symbol Δ denotes the ratio of the rotation rate aboutthe z axis to the rotation rate about the x axis, where only λ≤1 isconsidered herein. Thus, the pulse 402 rotates the Bloch vector aboutthe z axis by λ|θ|, which is less than the angle θ by the factor λ. Tominimize the rotation of the Bloch vector about the z axis, the pulseduration Δt should be made as short as possible.

At the nominal value Φ₀/2, the quantum-computational states |g

and |e

are separated by a nominal energy splitting ΔE that is equal to theLarmor frequency of the fluxonium qubit 120. As the magnetic flux Φdeviates from the nominal value Φ₀/2, the pulse 402 effectively acts asa transverse magnetic field that couples the states |g

and |e

, thereby increasing their energy splitting as the instantaneous flux Φincreasingly deviates away from the nominal value Φ₀/2 (see FIG. 11B).The energy splitting will achieve a maximum energy splitting at the peakof the pulse 402. In one embodiment, the amplitude A is selected suchthat the maximum energy splitting is at least twice the nominal energysplitting.

While FIG. 4 shows the magnetic flux having a nominal value ofapproximately Φ₀/2, the nominal value may be different than Φ₀/2. Infact, FIG. 9A shows that the relaxation time T₁ of the fluxonium qubit120 is smallest for magnetic fluxes near Φ₀/2. Accordingly, it may beadvantageous to operate at magnetic fluxes away from Φ₀/2. Operation ata magnetic flux away from Φ₀/2 will decrease the dephasing time T₂,which is largest at Φ₀/2 (see FIG. 9B). However, the increase in T₁ maybe greater than the reduction in T₂, giving rise to an optimal magneticflux at which the coherence properties of the fluxonium qubit 120 areoverall maximized.

While FIG. 4 shows the pulse 402 as being triangular with matched risingand falling edges (i.e., the slope of the rising edge equals thenegative of the slope of the falling edge). However, the pulse 402 mayhave any other shape without departing from the scope hereof. Forexample, the pulse 402 may have a “simple” shape that monotonicallydeviates away from the nominal value Φ₀/2 to a maximum deviation (eithergreater than or less than the nominal value), and then monotonicallydeviates from the maximum deviation back toward from the nominal valueΦ₀/2. The triangular pulse 402 shown in FIG. 4 is one example of asimple pulse shape that has no plateau (i.e., the pulse 402 does not“dwell” at any value of the flux). However, the pulse 402 mayalternatively include one or more plateaus. The pulse may deviatelinearly (e.g., rectangular, trapezoidal, triangular with unmatchedrising and falling edges, etc.) or nonlinearly (e.g., Gaussian,Lorentzian, etc.). Alternatively, the pulse 402 may have a “complex”shape that changes slope any number of times, with or without one ormore plateaus, before returning to the nominal value Φ₀/2. For example,the pulse 402 may also cross the nominal value Φ₀/2 one or more timesbefore returning to the nominal value Φ₀/2.

The pulse 402 can be advantageously generated using a commercialhigh-speed digital-to-analog converter (DAC), which is simpler toimplement and requires fewer components than prior-art techniques inwhich superconducting-qubit control signals are generated by mixing anenvelope function (e.g., as generated by an arbitrary waveformgenerator) with a high-frequency carrier. To increase stability, a DCpower supply can be used to continuously output a DC current thatgenerates the magnetic flux 130 at the nominal value. The DAC output maythen be AC-coupled to the DC current such that the pulses 402 deviatethe magnetic flux from the nominal value. Commercial DACs operate atspeeds greater than 10 Gbps, and therefore can achieve pulse durationsΔt less than 1 ns.

FIG. 5 is a plot of magnetic flux Φ as a function of time, illustratinga magnetic-pulse sequence 500 for universal single-qubit rotation of thefluxonium qubit 120. The magnetic-pulse sequence 500 uses two of thefast pulses 402 shown in FIG. 4 . The fluxonium qubit 120 may begin inany qubit state, i.e., any linear combination of thequantum-computational states |g

and |e

. Before an initial time t₁, the fluxonium qubit 120 is threaded withexternally generated magnetic flux 130 at a nominal value. Starting atthe initial time t₁, and lasting until a second time t₂=t₁+Δt₁, a firstpulse 402(1) is applied to the fluxonium qubit 120 by deviating themagnetic flux 130 in a first direction away from the nominal value. Inthe example of FIG. 5 , the magnetic flux Φ is increased above thenominal value Φ₀/2 by a first pulse amplitude A₁. The value Δt₁ is afirst pulse duration of the first pulse 402(1).

Starting at the second time t₂, and lasting until a third timet₃=t₂+Δt₁, the fluxonium qubit 120 idles with the magnetic flux Φ at thenominal value Φ₀/2 for an idling time Δt₁. Idling rotates the Blochvector by a second angle θ₂=Δt₁/ω_(q) about the z axis of the Blochsphere. The second angle θ₂ can be controlled by extending and/orshortening the idling time Δt₁.

Starting at the third time t₃, and lasting until a fourth timet₄=t₃+Δt₂, a second pulse 402(2) is applied by to the fluxonium qubit120 by deviating the magnetic flux 130 in a second direction, oppositethe first direction, away from the nominal value Φ₀/2. In the example ofFIG. 5 , the magnetic flux t is decreased below the nominal value Φ₀/2by a second pulse amplitude A₂. The value Δt₂ is a second pulse durationof the second pulse 402(2). The second pulse 402(2) rotates the Blochvector by the negative of the first angle (i.e., −θ₁) about the x axisof the Bloch sphere. Similar to the first pulse 402(1), the second pulse402(2) also rotates the Bloch vector about the z axis of the Blochsphere, due to the finite duration Δt₂, by λ|θ₁|. Note that the absolutevalue of θ₁ arises from the always-on rotation about the z axis.

The pulse amplitudes A₁, A₂ and pulse durations Δt₁, Δt₂, may beselected such that a first area of the first pulse 402(1) and a secondarea of the second pulse 402(2) sum to zero, wherein the magnetic-pulsesequence 500 is a zero-area pulse sequence. Here, the first and secondareas are measured relative to the nominal value Φ₀/2. Thus, in FIG. 5 ,the first pulse 402(1) has a positive area, and the second pulse 402(2)has a negative area. In one embodiment, A₁=−A₂ and Δt₁=Δt₂, as shown inFIG. 5 . However, the pulse amplitudes A₁, A₂ and pulse durations Δt₁,Δt₂ may have other values such that the first and second pulse areas sumto zero.

It may be beneficial to configure each pulse 402 to minimize high-orderharmonics (i.e., Fourier components of the sequence 500) that caninadvertently affect the behavior of the fluxonium qubit 120. To create“sharp” pulses 402, Fourier components will be needed at frequenciesabove 1/Δt_(s), where Δt_(s)=Δt₁+Δt₁+Δt₂ is a duration of the sequence500. However, at frequencies far above 1/Δt_(s) (e.g., ten times larger,or more), the amplitudes of the Fourier components may need to beattenuated. For this reason, triangular pulses 402 may be preferable torectangular pulses 402 since the Fourier spectrum of a triangular pulsetrain decreases faster with increasing frequency than that of arectangular pulse train.

As described in more detail below, values for each of the angles θ₁ andθ₂ may be selected, based on a given value of λ, such that themagnetic-pulse sequence 500 (i.e., the combined sequential effects ofthe first pulse 402(1), the idling, and the second pulse 402(2)) rotatesthe Bloch vector about they axis of the Bloch sphere by 90°. In thiscase, the magnetic-pulse sequence 500 can transform the fluxonium qubit120 from one of the pure states |g

and |e

into an equal superposition of the states |_(g)

and |e

(and vice versa). This version of the magnetic-pulse sequence 500 isfunctionally similar to a π/2 pulse, but can advantageously complete ina faster time. Accordingly, this version of the magnetic-pulse sequence500 can replace any π/2 pulse used in any quantum gate orquantum-computation algorithm.

Also described below, values for each of the angles θ₁ and θ₂ may alsobe selected, based on a given value of λ, such that the magnetic-pulsesequence 500 rotates the Bloch vector about the y axis of the Blochsphere by 180°. In this case, the magnetic-pulse sequence 500 cancoherently transfer the fluxonium qubit 120 between the states |g

and |e

. This version of the magnetic-pulse sequence 500 is functionallysimilar to a π pulse, but can advantageously complete in a faster time.Accordingly, this version of the magnetic-pulse sequence 500 can replaceany π pulse used in any quantum gate or quantum-computation algorithm(e.g., the π pulse 308, or the π pulse 310).

While FIG. 5 shows the first and second pulses 402(1), 402(2) astriangular, one or both of the first and second pulses 402(1), 402(2)may alternatively have a different shape without departing from thescope hereof. For example, one or both of the first and second pulses402(1), 402(2) may have a simple pulse shape (e.g., rectangular,trapezoidal, Gaussian, etc.) or a complex pulse shape, as describedabove for FIG. 4 .

While FIG. 5 shows the each of the pulse durations Δt₁, Δt₂ as beingless than the idling time t₁, one or both of the pulse durations Δt₁,Δt₂ may alternatively be greater than the idling time t₁. For example,each of the pulse durations Δt₁, Δt₁ may equal twice the idling time t₁,as may occur when each of the first and second pulses 402(1), 402(2) isrectangular. In some embodiments, there is no idling between the firstand second pulses 402(1), 402(2), i.e., the second pulse 402(2) beginswhen the first pulse 402(1) ends, or t₁=0.

In one embodiment, the magnetic-pulse sequence 500 is applied twice tothe fluxonium qubit 120 to rotate the Bloch vector by an arbitrary angleϕ about the x axis of the Bloch sphere. In a first sequence 500, theangles θ₁ and θ₂ are selected such that the first sequence 500 rotatesthe Bloch vector by −90° rotation about the y axis of the Bloch sphere.After the first sequence 500, the magnetic flux idles at the nominalvalue (e.g., Φ₀/2), which rotates the Bloch vector by the arbitraryangle ϕ about the z axis of the Bloch sphere. After the idling, a secondsequence 500 rotates the Bloch vector by +90° about the y axis of theBloch sphere.

FIG. 6 shows one example of a complex pulse that crosses the nominalflux several times. This pulse rotates the fluxonium qubit 120 butwithout idling. Accordingly, in an embodiment, the magnetic-pulsesequence 500 is a complex pulse that crosses the nominal flux one ormore times, with or without idling.

In some embodiments, a method for manipulating a fluxonium qubit stateincludes applying a flux pulse (e.g., the pulse 402 of FIG. 4 ) to thefluxonium (e.g., the fluxonium qubit 120 of FIG. 1 ). The fluxonium maybe biased at or near a flux of one-half the superconducting magneticflux quantum Φ₀, i.e., Φ₀/2. Alternatively, the fluxonium may be biasedaway from Φ₀/2. Said manipulating may occur in less than N periods ofthe Larmor oscillation (e.g., N=2 periods or N=10 periods). The fluxpulse may include any one or more of the following properties: (1)starts and stops at the same flux (which gives composability); (2) has anet zero area (cuts of sensitivity to low frequency line filtering,etc.); (3) first- and/or second-order order insensitivity tolow-frequency flux noise; (4) first- and/or second-order insensitivityto small changes of tunnel splitting; (5) first- and/or second-orderinsensitivity to small changes in the flux bias; (6) is filtered and/orgenerated with limited power contained in high frequencies for a givenlength of pulse (makes it easier to calibrate lines); (7) is filteredand/or generated with little power at frequencies near other transitionsin the fluxonium (i.e., no leakage).

In some embodiments, a measurement method for determining a fluxoniumqubit state may operate in a z basis (i.e., symmetric/anti-symmetric),wherein the fluxonium is biased at or near a flux of one-half thesuperconducting magnetic flux quantum Φ₀ (i.e., Φ₀/2). Alternatively,the measurement method may operate in or an x basis (which well),wherein the fluxonium is biased away from Φ₀/2. In either case, themethod includes performing a quantum non-demolition (QND) measurement onthe fluxonium qubit state in the corresponding basis. A duration of theQND measurement may be shorter than the coherence time of the fluxonium.The QND measurement may also be “latching”, i.e., is robust againstqubit state changes during the measurement. When the QND measurement islatching, the duration of the QND measurement may be longer than thequbit coherence time.

The measurement method may use dispersive coupling of a fluxon state ofthe fluxonium to a structured radiation environment (e.g., a resonator,filter, cavity, etc.). In this case, active pulses may be used to probethe state-dependent change of the structured radiation environment. Themeasurement method may use shaped pulses. The measurement method mayalso use dispersive coupling of a plasmon state of the fluxonium to thestructured radiation environment using one or more of the following: (1)direct occupation of the plasmon state; (2) virtual coupling to theplasmon state; (3) one or more active pulses to probe thestate-dependent change of the structured radiation environment; (4)emission of energy of the plasmon state into a measurement apparatus;(5) one or more active pulses to probe the qubit state-dependence ofplasmons; and (6) one or more shaped pulses. In some embodiments, ameasurement method includes the manipulation method described above.

In some embodiments, an initialization method for cooling a fluxonium toa known state may use plasmons. For example, the initialization methodmay use one or more of plasmon dissipation, direct excitation ofplasmons, and virtual excitation of plasmons to mediate coupling to acold bath. The initialization method may also use radiative cooling. Forexample, the initialization method may implement radiative cooling usingone or more of a transmission line, a structured radiation environment(e.g., a resonator, a bandpass filter, a high pass filter, a low-passfilter, etc.). Alternatively, the initialization method may use acombination of plasmons radiative cooling. The fluxonium (or anothertype of superconducting qubit) has a transition energy between itsquantum-computational states that is small enough such that there is asignificant equilibrium population of each of the quantum-computationstates (i.e., hω<k_(B)T, where ℏ is Planck's constant divided by 2π,k_(B) is Boltzmann's constant, ω is the angular frequency correspondingto the transition energy, and T is the temperature of the environment).In one embodiment, hω<5k_(B)T. In another embodiment, hω≤k_(B)T.

In some embodiments, the initialization method uses the manipulationmethod(s) described above. In some embodiments, the initializationmethod uses the manipulation method(s) as well as the measurementmethod(s) described above.

In some embodiments, the measurement method is combined with theinitialization method (e.g., the initialization method may be performedprior to the measurement method). Similarly, the initialization methodmay be combined with one or both of the measurement method and themanipulation method.

Experimental Demonstration

Introduction

Superconducting circuits are among the fastest developing candidates forquantum computers due to steady improvements in coherence times, gatefidelities, and processor size. These developments have ushered thenoisy intermediate-scale quantum era and demonstrations of quantumadvantage over classical computing. Many superconducting quantumprocessors are based on the transmon circuit, which since its inceptionhas seen improvements in coherence by nearly 4-5 orders of magnitudedriven largely by decreasing environmental noise. While the transmoncircuit has seen widespread use in quantum computation, fluxonium offersmany advantages over earlier flux qubits, including a rich levelstructure, natural protection from charge-noise induced relaxation anddephasing, and reduced sensitivity to flux noise. One of the challengesin making fluxonium a building block for superconducting qubitprocessors arises from the slow gates using standard microwave control.

In this section, we demonstrate high-fidelity control of a fluxoniumcircuit using a universal set of single-cycle flux gates on a qubitwhose frequency is an order of magnitude lower than the ambienttemperature. In the process, we reimagine all aspects of how the circuitshould be controlled and operated, and demonstrate coherence times andgate fidelities that match or exceed those of the best transmoncircuits, with the potential for further improvements.

The transmon is one of the simplest in the family of superconductingcircuits, realizing a weakly anharmonic oscillator with large dipolematrix elements. This circuit trades off increased sensitivity to decay,and a reduced anharmonicity for decreased sensitivity tocharge-noise-induced dephasing. Despite the maximal susceptibility torelaxation, state-of-the-art transmons have relaxation (T₁) times around100 μs, corresponding to Qs of a few million. The gate speeds are,however, limited by the small anharmonicity, typically ˜5% of the qubitfrequency ω_(q), resulting in a theoretical upper bound of˜ω_(q)/(Qα)˜10⁻⁵ and state-of-the-art values of ≲1-2×10⁻⁴ for the gateinfidelity. This suggests that gate fidelities can be made to approach1/Q by increasing the anharmonicity in comparison to the qubit frequencyω_(q), and performing gate operations within a few Larmor periods.

The flux qubit, another member of the superconducting circuit family,already has the desired level structure with a relative anharmonicityα/ω_(q)>>1. The extreme sensitivity to flux noise of these qubits wasmitigated by shunting the Josephson junction with a large superinductor,resulting in the development of the fluxonium. Further improvements inenergy relaxation times were obtained by the realization of a heavyfluxonium, which additionally reduced the decay matrix elements using alarge shunting capacitor. These variants of fluxonium are reported tohave longer coherence times than transmons in 3D architectures. Eventhough heavy fluxonium has the desired level structure and largecoherence times, fast manipulation of the metastable qubit statesremains a challenge due to the suppressed charge matrix elements. WhileRaman transitions can be used for coherent operations, these protocolsare still relatively slow and require high drive powers, while exposingthe qubit to the higher loss rates of excited fluxonium levels involvedduring the gate.

In this work, we realize a heavy-fluxonium circuit in a 2D architecturewith coherence times T₁, T_(2e)˜300 μs exceeding those of standardtransmons. The frequency of the qubit transition is only 14 MHz, anorder of magnitude lower than the temperature of the surrounding bath.Therefore, to initialize the qubit we develop and realize a resetprotocol that utilizes the readout resonator and higher circuit levelsto initialize the qubit with 97% fidelity, effectively cooling the qubitdown to 190 μK. Lastly, we use flux pulses to realize high-fidelitysingle-qubit gates within a single period 2π/ω_(q) of the Larmoroscillation.

The Heavy-Fluxonium Circuit

The circuit consists of a small-area Josephson junction (JJ) withinductance L_(J) shunted by a large inductance (L_(JA)), and a largecapacitor (C_(q)), as shown in FIG. 7A. The shunting inductance isrealized by an array of 300 large-area JJs, each having a Josephsonenergy E_(JA) and charging energy E_(CA). We make E_(JA)/E_(CA)>>1 toensure that the charge dispersion for each array junctions is small, andthe array can be regarded as a linear inductor. The correspondingeffective circuit is shown in FIG. 7B, and the resulting Hamiltonian isgiven by Eqn. 1. The corresponding values for the reported device are:E_(C)/h=0.479 GHz, E_(L)/h=0.132 GHz, and E_(J)/h=3.395 GHz, where h isPlanck's constant. The level structure of the fluxonium at theflux-frustration point (Φ_(ext)=Φ₀/2) is shown in FIG. 7C. There are twotypes of transitions of interest, the intra-well plasmons (|g

↔h

and |e

↔|f

) and the inter-well fluxons (|g

↔|e

and |f

↔|h

). The single-photon transitions |g

↔|f

and |e

↔|h

are forbidden at the flux-frustration point due to the parity selectionrule. The qubit is comprised of the lowest two energy levels |g

and |e

, with the qubit transition being fluxon like, with a frequency ω_(q) of14 MHz.

Qubit Initialization and Readout

Due to its low transition frequency, the qubit starts in a nearlyevenly-mixed state in thermal equilibrium. We first initialize the qubitin a pure state (|g

or |e

) using the reset protocol shown in FIG. 8A. In this protocol, wesimultaneously drive both the |g0

→|h0

and |h0

→|e1

transitions for 15 μs. The high resonator frequency (5.7 GHz) incomparison to the physical temperature, and the low resonator qualityfactor Q=600 result in the rapid loss of a photon from |e1

, effectively removing the entropy from the qubit. In conjunction withthe large matrix element between |h0

and |e1

, this steers the system into a steady state with over 95% of thepopulation settling in |e0

in 5 μs. We subsequently perform an additional π pulse on the |g

-|e

transition to initialize the system in the ground state (|g0

). The reset is characterized by performing a Rabi rotation between the|e

↔|f

levels, as shown in FIG. 2(b). The Rabi contrast is doubled followingreset, consistent with 50% of the population being in |e

in thermal equilibrium. If we prepare the system in |g

, the |e

χ|f

Rabi contrast indicates a 3±2% error in state preparation, depending onthe |f

state thermal population. Since the |f

frequency is similar to the typical transmon frequencies, its thermalpopulation is in line with that of most transmons. The effective qubittemperature following reset is ˜190 μK, lower than the ambienttemperature by a factor of 100.

Readout of the fluxonium levels is performed using circuit quantumelectrodynamics by capacitively coupling the fluxonium circuit to areadout resonator. Since the qubit states are far away in frequency fromthe readout resonator, the dispersive shift x of the resonator due to achange in the occupation of computational states is small (60 kHz).While the large detuning reduces the qubit heating through theresonator, it makes direct dispersive readout challenging. We circumventthis issue by utilizing the larger dispersive interactions χ_(f), χ_(h)of the excited levels |f

, |h

, which are closer in frequency to the readout resonator. To improvereadout fidelity, we thus perform a π pulse on the |e

-|f

transition in 80 ns, before standard dispersive readout. Since thepopulation in |e

is transferred to |f

, the readout signal becomes proportional to (χ_(f)-χ_(g)), which isfive times larger than (χ_(e)-χ_(g)). This plasmon-assisted readoutscheme results in 50% single-shot readout fidelity, which can be furtherimproved with a parametric amplifier, and by optimizing the resonator Kand the dispersive shifts.

Characterizing Device Coherence

Having developed protocols for initialization and readout, wecharacterize the coherence properties of the qubit. The inset of FIG. 9Ashows a T₁=315±10 pts measured at the flux-frustration point followinginitialization of the qubit in either the |g

or |e

state. The qubit relaxes to a near equal mixture where the excited statepopulation P(|e

)=0.4955±0.0015, with the deviation providing an estimate of thetemperature of the surrounding bath, T=42±14 mK. At the flux-frustrationpoint, the wavefunctions are delocalized into symmetric andanti-symmetric combinations of the states in each well. As we move awayfrom this degeneracy point, the wavefunctions localize into differentwells resulting in a suppression of tunneling and an increase in therelaxation times, see FIG. 9A. Here, the qubit relaxation times weremeasured over a wide range of external flux by driving the |g

-|h

transition for 120 μs to pump the qubit into the |e

state, and monitoring the subsequent decay. While moving away from theflux-frustration point, T₁ increases to a maximum value of 4.3±0.2 ms,consistent with previous heavy-fluxonium devices, before subsequentlydecreasing.

To explain the measured relaxation times, we consider several avenues bywhich the qubit can decay, including Purcell loss, decay via charge andflux coupling to the control lines, 1/f flux noise, dielectric loss inthe capacitor, and resistive loss in the superinductor. Conservativeestimates of the flux noise induced loss are lower than the measuredloss by nearly an order of magnitude. The loss near the flux-frustrationpoint is believed to be largely due to dielectric loss in the capacitor.This can be thought of as Johnson-Nyquist current noise from theresistive part of the shunting capacitor, which couples to the phasematrix element

g|{circumflex over (φ)}|e

, and grows rapidly as we approach the flux-frustration point. Assuminga fixed loss tangent for the capacitor, this loss rate is inverselyproportional to the impedance of the capacitor, and is given by:

$\begin{matrix}{\Gamma_{diel} = {\frac{\hslash\omega_{q}^{2}}{8E_{C}Q_{cap}}{\coth\left( \frac{{\hslash\omega}_{q}}{2k_{B}T} \right)}{{❘\left\langle {g{❘\hat{\phi}❘}e} \right\rangle ❘}^{2}.}}} & (2)\end{matrix}$

The T₁ at the flux-frustration spot sets an upper bound of1/Q_(cap)=8×10⁻⁶ for the loss tangent of the capacitor, which is withina factor of three of the value reported in previous heavy-fluxoniumdevices, and results in the dashed curve in FIG. 9A. Since ω_(q) isbelow the ambient temperature near the flux-frustration point, acombination of the temperature-dependent prefactor ˜2k_(B)T/(ℏω_(q)),and the relation between charge and phase matrix elements in fluxonium,

g|{circumflex over (n)}|e

=ω/(8E_(c))

g|{circumflex over (ϕ)}|e

, results in the dielectric-loss scaling as 1/ω, which is consistentwith the observed trend in the T₁ near the flux-frustration point. Themeasured T₁ at the flux-frustration point also sets an upper bound of5×10⁻⁹ for the loss tangent of the inductor. The decay from inductiveloss, however, increases more rapidly with frequency than dielectricloss (∝1/ω³) and is inconsistent with measured data. Our qubitoperations are performed between 0.4Φ₀-0.5Φ₀ where the T₁ is mainlylimited by dielectric loss. As we move further away from theflux-frustration point (˜0.4Φ₀), T₁ starts to decrease. This additionalloss is believed to be due to a combination of radiative loss to thecharge drive line, and Purcell loss from higher fluxonium levels excitedby heating from the |g

and |e

states. The Purcell loss calculated based on the coupledfluxonium-resonator system using a bath temperature of 60 mK results inthe dotted blue curve shown in FIG. 9A. The enhanced loss nearΦ_(ext)=0.35Φ₀ is suggestive that heating to higher levels maycontribute as there are several near resonances of higher fluxoniumlevels with the readout resonator, which depend sensitively on thecircuit parameters.

The dephasing is characterized using a Ramsey sequence with three echo πpulses, and found to be minimized at Φ_(ext)=Φ₀/2, where the qubitfrequency is first-order insensitive to changes in flux. The dephasingrate near the flux-frustration point can be separated into two parts.The first is a frequency-independent term Γ_(C) mainly composed of qubitdepolarization, and dephasing from cavity photon shot noise and otherflux insensitive white noise sources. The second arises from 1/f fluxnoise that is proportional to the flux slope as

${\Gamma_{1/f} = {\frac{d\omega}{d\phi}\eta\sqrt{W}}},$

where η is in the flux-noise amplitude and W depends on the number of πpulses in an echo experiment (W=4 ln 2−9/4 ln 3 for three π pulses).Thus, our spin-echo signal decays as exp(−t/T_(C))×exp(−Γ_(1/f) ²t²).Here T_(C)=1/Γ_(C) is the T_(2e) value at the flux-frustration point. Itis found to be ˜300 μs, much higher than the T_(2e) values forstate-of-the-art transmons, see inset of FIG. 9B. The T_(2e) valuesaround the flux-frustration point, defined as the time for the echooscillation amplitude to decay to 1/e are shown in FIG. 9B. This valuefalls off rapidly as we move away from the flux-frustration point,consistent with the small tunnel coupling between levels. Away from theflux-frustration point, T_(2e) is mainly limited by 1/f flux noise. TheT_(2e) far from the frustration point is projected to be ˜10 μsaccording to our model, which is consistent with other reported results.

Fast Single-Cycle Flux Gates

To maximize the advantage of the large anharmonicity of heavy fluxonium,we rethink the standard microwave-drive control of the circuit which ishindered by the suppressed charge matrix elements. We instead performhigh-fidelity gates through fast flux pulses, similar to the controlscheme used in the original charge qubit. Near the flux-frustrationpoint where the fluxonium is operated, the Hamiltonian within thecomputational space can be idealized as a spin-½ system,

$\frac{H}{h} = {{\frac{A\left( \Phi_{ext} \right)}{2}\sigma_{x}} + {\frac{\Delta}{2}{\sigma_{z}.}}}$

Here Δ≈14 MHz is the splitting of |g

and |e

at the flux-frustration point, and corresponds to the qubit frequencyω_(q). The amplitude of the σ_(x) term is proportional to the fluxoffset δΦ_(ext) from the flux-frustration point, and given by A=4π

g|{circumflex over (φ)}|e

E_(L)δΦ_(ext)/h. The coefficient of the σ_(x) term can be much largerthan the qubit frequency, with A˜300 MHz when δΦ_(ext)=0.06Φ₀,disallowing any rotating wave approximation.

FIG. 10A shows the protocol for a generic qubit pulse. We first rapidlymove the flux-bias point away from the flux-frustration point in onedirection and back, thus generating a rotation about the x axis througha large σ_(x) term in our computational basis. There is additionally arelatively small rotation about the z axis corresponding to the timeΔt_(p) of the triangular spike. We subsequently idle at theflux-frustration point for a duration Δt_(z), which results in arotation by ω_(q)Δt_(z) about the z axis. Finally, we rapidly move theflux-bias point in the other direction and back, resulting in a −σ_(x)term and another small z rotation. We choose the two spikes to beexactly anti-symmetric, ensuring zero net flux, simultaneouslyminimizing the effect of microsecond and millisecond pulse distortionsubiquitous in flux-bias lines, and echoing out low-frequency noise. Thepulse is also immune to shape distortions since the total σ_(x) andσ_(z) amplitudes depend only on the area of the spike and Δt_(z). Bysweeping the amplitude A of the triangular spike and idling lengthΔt_(z) of the pulse, and measuring the expectation value of the spinalong each axis, we obtain the 2D Rabi patterns shown in FIG. 10C thatprovide a measure of our gate parameters. A vertical line cut of thesegraphs corresponds to Larmor precession in the lab frame, with anoscillation frequency of Δ=14 MHz. We thus obtain a Z/2 gate by idlingat the flux-frustration point for Δt_(z)=1/(4Δ). We obtain a Y/2 gate atthe point indicated by a star 1002, with the corresponding trajectorieson the Bloch sphere for three different cardinal states shown in FIG.10D. Y/2 and arbitrary rotations about the z axis are sufficient foruniversal control. An X/2 gate, for instance, is performed through thecombination (−Y/2)·(Z/2)·(Y/2).

We characterized the fidelities of our single-qubit gates throughrandomized benchmarking (RB) and interleaved RB (IRB). RB provides ameasure of the average fidelity of single-qubit Clifford gates and isperformed by applying sequences containing varying number of Cliffordgates on the state |e

. For a given sequence length, we performed 75 randomized sequences,each containing a recovery gate to the state |e

before the final measurement. IRB allowed us to isolate the fidelitiesof individual computational gates and was performed by interleaving thegate between the random Clifford gates of the RB sequence. The averageddecay curves of P(|e

) as a function of the sequence length for standard RB (black circles),and IRB for Z/2 (triangles), Y/2 (diamonds) and X/2 (squares) gates areshown in FIG. 10E. The infidelities thus extracted for the Y/2, Z/2, andX/2 gates are 8, 1, and 24×10⁻⁴, respectively. The X/2 gate infidelityis slightly worse than the combined infidelities from two Y/2, and oneZ/2 gate. The durations for Y/2 and Z/2 are ˜20 ns, while that for theX/2 gate is ˜60 ns, and thus all the computational gates are performedwithin one qubit Larmor period 2π/ω_(q)=70 ns, with all the operationsoccurring in the lab frame. The calculated decoherence limited errors ofthe Y/2, and X/2 gates are 6.67×10⁻⁵ and 2×10⁻⁴, suggesting that themajor source of gate error arises from residual calibration errors inthe pulse parameters, providing room for improvement even from thesestate-of-the-art values.

Experimental Setup

The experiment was performed in a Bluefors LD-250 dilution refrigeratorwith the wiring configured as shown in FIG. 11 . The flux and chargeinputs are attenuated with standard XMA attenuators, except the final 20dB attenuator on the RF charge line (threaded copper). The DC andRF-flux signals were combined in a modified bias-tee (Mini-CircuitsZFBT-4R2GW+), with the capacitor replaced with a short. The DC andRF-flux lines included commercial low-pass filters (Mini-Circuits) asindicated. The RF flux and output lines also had additional low-passfilters with a sharp cutoff (8 GHz) from K&L microwave. Eccosorb (CR110)IR filters were added on the flux, and output lines, which helpedimprove the T₁ and T₂ times, and reduce the qubit and resonatortemperatures. The device was heat sunk to the base stage of therefrigerator (stabilized at 15 mK) via an OFHC copper post, whilesurrounded by an inner lead shield thermalized via a welded copper ring.This was additionally surrounded by two cylindrical μ-metal cans(MuShield), thermally anchored using an inner close fit copper shimsheet, attached to the copper can lid. We ensured that the sample shieldwas light tight, to reduce thermal photons from the environment.

Device Fabrication

The device (see FIG. 7A) was fabricated on a 430 μm thick C-planesapphire substrate. The base layer of the device, which includes themajority of the circuit (excluding the Josephson junctions), consists of150 nm of niobium deposited via electron-beam evaporation, with featuresfabricated via optical lithography and reactive ion etch (RIE) atwafer-scale. 600 nm thick layer of AZ MiR 703 was used as the (positive)photoresist, and the large features were written using a Heidelberg MLA150 Direct Writer, followed by RIE performed using a PlasmaTherm ICPFluorine Etch tool. The junction mask was fabricated via electron-beamlithography with a bi-layer resist (MMA-PMMA) comprising of MMA EL11 and950PMMA A7. The e-beam lithography was performed on a Raith EBPG5000Plus E-Beam Writer. All Josephson junctions were made with the Dolanbridge technique. They were subsequently evaporated in Plassys electronbeam Evaporator with double angle evaporation) (±19°. The wafer was thendiced into 7×7 mm chips, mounted on a printed circuit board, andsubsequently wire-bonded.

Deconstruction of Single-Qubit Gates

Modulation of the external flux drive with appropriate amplitude andduration is sufficient to perform arbitrary single-qubit rotations. Thenative gates available in our system are the arbitrary phase gateR_(z)(θ) which rotates the qubit by an arbitrary angle θ about theZ-axis and a combination of X- and Z-rotation R_(xz)(θ). R_(z)(θ) isrealized by waiting for a period of Δt_(z)=θ/ω_(q) (since we are workingin the lab frame) whereas R_(xz)(θ) is implemented by a flux-driveapplied for a duration of Δt_(p)=λθ/ω_(q). Here λ (λ≤1) is the ratio ofZ-rotation to X-rotation rates. These rotation matrices can be expressedas,

$\begin{matrix}{{{R_{z}(\theta)} = e^{{- i}\sigma_{z}\theta/2}},} & ({C1})\end{matrix}$ $\begin{matrix}{{R_{xz}(\theta)} = {e^{{- {i({{\theta\sigma_{x}} + {\lambda{❘\theta ❘}\sigma_{z}}})}}/2}.}} & ({C2})\end{matrix}$ $\begin{matrix}{{{R_{z}(\theta)} = e^{{- i}\sigma_{z}\theta/2}},} & \text{?}\end{matrix}$ ?indicates text missing or illegible when filed

The |θ| in Eqn. C2 arises due to the always-on Z-rotation which isunidirectional in the lab frame. A generic zero-flux-pulse can beconstructed as,

(θ)=R _(xz)(−θ_(x))·R _(z)(θ_(z))·R _(xz)(θ_(x))  (C3)

A π/2 rotation about the Y-axis (Y/2), i.e.,

$\begin{matrix}{{{R_{y}\left( {\pi/2} \right)} = {\frac{1}{\sqrt{2}}\begin{pmatrix}1 & {- 1} \\1 & 1\end{pmatrix}}},} & ({C4})\end{matrix}$

is obtained using

$\begin{matrix}{{\theta_{x} = {\frac{1}{\sqrt{1 + \lambda^{2}}}{\cos^{- 1}\left\lbrack \frac{\lambda\left( {1 + \lambda} \right)}{- \left( {1 - \lambda} \right)} \right\rbrack}}},} & ({C5A})\end{matrix}$ $\begin{matrix}{\theta_{z} = {2{\tan^{- 1}\left\lbrack \frac{\sqrt{1 - {2\lambda} - {2\lambda^{3}} - \lambda^{4}}}{\left( {1 + \lambda} \right)\sqrt{1 + \lambda^{2}}} \right\rbrack}}} & ({C5B})\end{matrix}$

in Eqn. C3 provided 0≤λ≤√{square root over (2)}−1. Similarly, we canconstruct

$\begin{matrix}{{R_{y}(\pi)} = {\begin{pmatrix}0 & {- 1} \\1 & 0\end{pmatrix} = {{- i}\sigma_{y}}}} & ({C6})\end{matrix}$

using

$\begin{matrix}{{\theta_{x} = {\frac{1}{\sqrt{1 + \lambda^{2}}}{\cos^{- 1}\left( \lambda^{2} \right)}}},} & ({C7A})\end{matrix}$ $\begin{matrix}{{\theta_{z} = {\pi - {2{\tan^{- 1}\left\lbrack \frac{\lambda}{\sqrt{1 - \lambda^{2}}} \right\rbrack}}}},} & ({C7B})\end{matrix}$

with 0≤λ≤1. An arbitrary rotation about X-axis can be constructed using

R _(x)(θ)=R _(y)(π/2)·R _(z)(θ)·R _(y)(−π/2).  (C8)

These gates are sufficient to construct any single-qubit unitaryoperation. We used the QuTiP python package to simulate the evolution ofthe computational levels under application of the pulse shown in FIGS.9A and 9B, and obtained the gate parameters. We swept the driveamplitude A and idling period Δt_(z) in our simulation to match thesweep performed in the experiment, as shown in FIG. 12 . For allexperiments and simulations reported herein, Δt_(p)=4.76 ns.

Clifford Gate Lengths and Fidelities

A complete Clifford set includes the computational gates(exp(±iπσ_(j)/4), j=x, y) and the Pauli gates (exp(±iπσ_(j)/2), j=I, x,y, z). In this work, we constructed Y/2 and Z/2 gates, and used them asbuilding blocks for the other gates in the Clifford Set. The total gatelengths, experimental infidelities (computational gates only), and gatecompositions are shown in Table 1. The computational gate lengths rangefrom 21-60 ns, and the longest Pauli gate (X) has a length of 78 ns.Since 2π/ω_(q)≈70 ns, the computational gates are all within a singlecycle of the qubit, and the longest gate is around one cycle as well.The microwave driving gates have lengths longer than ˜10×2π/ω_(q), soour gates are 10 to 30 times faster.

TABLE 1 Clifford Gate lengths and fidelities Gate Length (ns)Experimental Fidelity Gate Composition Y/2 21.19  8 × 10⁻⁴ Z/2 17.87  1× 10⁻⁴ X/2 60.25 24 × 10⁻⁴ Y/2, Z/2, −Y/2 Y 42.38 Y/2, Y/2 Z 35.73 Z/2,Z/2 X 78.11 Y/2, Z, −Y/2

Fluxonium Matrix Elements and Reset Protocol

We derive the charge drive transition rates by simulating the fullqubit-resonator dressed system. The drive power is normalized to 258 MHzso that the |g0

→|h0

π pulse takes 80 ns, which corresponds to the typical experimentalvalue. The simulated single-photon and two-photon transition rates (inMHz) are shown in Tables 2 and 3, respectively. The observed transitionrates have additional contributions arising from the frequencydependence of the transmission through the drive line.

TABLE 2 One-Photon Matrix Elements |g0 

|e0 

|f0 

|h0 

|g1 

|e1 

|g0 

0.0738 6.2577 257.9425 |e0 

0.0738 5.8679 257.9425 |f0 

5.8679 1.2475 0.0138 |h0 

6.2577 1.2475 0.1028 |g1 

257.9425 0.0138 0.0741 |e1 

257.9425 0.1028 0.0741

TABLE 3 Two-Photon Matrix Elements |g0 

|e0 

|f0 

|h0 

|g1 

|e1 

|g0 

1.9213 0.9177 |e0 

1.6489 0.4207 |f0 

1.9213 0.0644 |h0 

1.6489 0.1258 |g1 

0.4207 0.1258 |e1 

0.9177 0.0644

We utilized the |g0

→|h0

and |h0

→|e1

transitions for the reset protocol due their large matrix elements.While the |g0

→|e1

two-photon process also has a relatively high rate, its use results indeleterious consequences since it lies in the middle of othertransitions. The excited state population as a function of reset time isshown in FIG. 13 . The majority of the population is pumped to state |e

in 5 μs, which is mainly determined by the |h0

→|e1

transition rate. We subsequently perform an additional π pulse on the |g

-|e

transition to initialize the system in the ground state |g0

.

Plasmon-Assisted Readout

The resonator frequency shifts in increasing order are χ_(e), χ_(g),χ_(h), χ_(f). We selected the |g

, |f

states for plasmon-assisted readout since χ_(f)-χ_(g) is larger thanχ_(h)-χ_(e). This is reflected in the single-shot readout histogram datafor |g

, |e

, |f

, |h

as shown in FIGS. 14A and 14B. The histograms in FIG. 14A are not wellseparated since the current sample is not optimized for high-fidelityreadout.

Modeling Fluxonium Relaxation

To explain the measured relaxation times of the fluxonium, we considerdecay via charge and flux coupling to the control lines, 1/f flux noise,dielectric loss in the capacitor, resistive loss in the superinductor,and Purcell loss. The decay rates arising from these loss mechanisms arederived using Fermi's golden rule, with the bath described using theCaldeira-Leggett model. For a noise source with amplitude f(t) andcoupling constant α between the fluxonium qubit states, the interactionHamiltonian can be written as H′=αf(t)σ_(x) in the qubit subspace. Thisresults in a qubit depolarization rate,

$\begin{matrix}{\Gamma = {\frac{a^{2}}{\hslash^{2}}{\left( {{S_{f}\left( {+ \omega_{01}} \right)} + {S_{f}\left( {- \omega_{01}} \right)}} \right).}}} & \left( {G1} \right)\end{matrix}$

Here S_(f)(ω)=∫_(−∞) ^(∞)e^(iωτ)

<f(τ)f(0)

is the noise spectral density associated with the source. We note thatat a finite bath temperature corresponding to an inverse temperature

${\beta = \frac{1}{k_{B}T}},$

detailed balance relates the positive and negative frequency componentsof the noise spectral density as S_(f)(−ω)/S_(f)(ω)=e^(−βℏω). Dependingon the noise source f, the coupling constant α is proportional to thecharge or phase matrix element of the fluxonium. Since the only term inthe Hamiltonian that does not commute with {circumflex over (ϕ)} is thecharging energy 4E_(c){circumflex over (n)}², and [{circumflex over(ϕ)}, {circumflex over (n)}]=i,

$\begin{matrix}{\ \begin{matrix}{\left\langle {j{❘\left\lbrack {\overset{\hat{}}{\phi},\overset{\hat{}}{H}} \right\rbrack ❘}k} \right\rangle = {\left( {\omega_{j} - \omega_{k}} \right)\left\langle {j{❘\overset{\hat{}}{\phi}❘}k} \right\rangle}} \\{= {{i\left( {8E_{c}} \right)}{\left\langle {j{❘\overset{\hat{}}{n}❘}k} \right\rangle.}}}\end{matrix}} & \left( {G2} \right)\end{matrix}$

The matrix elements of the fluxonium circuit are thus related by

${❘\left\langle {g0} \middle| \overset{\hat{}}{n} \middle| {g1} \right\rangle ❘} = {\left( \frac{\omega}{8E_{c}} \right){❘\left\langle {g0{❘\overset{\hat{}}{\phi}❘}g1} \right\rangle ❘}}$

for all flux values.

Relaxation from Flux Noise

Flux noise couples to the phase degree of freedom with an interactionstrength that depends on the inductive energy E_(L). Expanding thefluxonium potential to lowest order in flux results in a couplingconstant of α=2πE_(L)

g0|{circumflex over (φ)}|g1

/Φ₀. We consider flux noise contributions from current noise in theflux-bias line, as well as 1/f flux noise. In our experimental setup,the current noise is believed to be mainly due to resistiveJohnson-Nyquist noise arising from a 10-dB attenuator with resistanceR=26Ω (last resistor in T network) on the fast flux line, correspondingto current noise spectral density of

${{S_{I}(\omega)} = {\frac{2}{R}\frac{\hslash\omega}{\left( {1 - e^{{- \beta}\hslash\omega}} \right)}}},$

with the expected interpolation between quantum and thermal noise. Thisis related to flux noise by the mutual inductance M=θ₀/1.6 mA betweenflux line and the qubit, obtained from the DC flux period. Therefore,

${{{S_{f}(\omega)} + {S_{f}\left( {- \omega} \right)}} = {2\hslash\omega\frac{M^{2}}{R}{\coth\left( \frac{\beta\hslash\omega}{2} \right)}}},$

and the decay rate

$\begin{matrix}{{\Gamma_{R} = \left. {{\pi^{3}\left( \frac{R_{Q}}{R} \right)}\left( \frac{M}{L} \right)^{2}\left\langle {g0{❘\overset{\hat{}}{\varphi}❘}g1} \right\rangle} \middle| {}_{2}{\omega{\coth\left( \frac{\beta\hslash\omega}{2} \right)}} \right.},} & \left( {G3} \right)\end{matrix}$

where R_(Q)=h/e² is the resistance quantum, and L is the fluxoniuminductance.

For 1/f flux noise, the noise spectral density is of the formS_(ϕ)(ω)=2πη²/ω, with the resulting decay rate,

$\begin{matrix}{\Gamma_{1/f} = {8{\pi^{3}\left( \frac{E_{L}}{\hslash} \right)}^{2}\left( \frac{\eta}{\Phi_{0}} \right)^{2}{\frac{{\left. {{❘\left\langle {g0} \right.❘}\overset{\hat{}}{\varphi}{❘{g1}}} \right\rangle ❘}^{2}}{\omega}.}}} & \left( {G4} \right)\end{matrix}$

The 1/f noise amplitude is fit from T_(2e) data, and corresponds toη=5.21μΦ₀. The suppression of the 1/f noise induced decay by E_(L) ²,results in a limit of T₁=2.4 ms for the relaxation time at theflux-frustration point, which grows rapidly (∝ω³) as we move away fromit.

Relaxation from Radiation Loss to the Charge Line

In addition to current noise, the fluxonium could also be affected byradiative loss arising from Johnson-Nyquist voltage noise

$\left( {{S_{V}(\omega)} = \frac{2R\hslash\omega}{1 - e^{{- \beta}\hslash\omega}}} \right)$

that couples to the qubit via spurious charge coupling, with theresistance R serving as a phenomenological parameter. In this case, thecoupling constant is related to the charge matrix element as α=2e

g0|{circumflex over (n)}|g1

, and

${{{S_{f}(\omega)} + {S_{f}\left( {- \omega} \right)}} = {2R\hslash\omega{\coth\left( \frac{\beta\hslash\omega}{2} \right)}}}.$

The resulting decay rate is

$\begin{matrix}{{\left. {\Gamma_{c} = {\frac{\omega}{Q_{c}}{\coth\left( \frac{\beta\hslash\omega}{2} \right)}{❘\left\langle {g0} \right.❘}\overset{\hat{}}{n}{❘{g1}}}} \right\rangle ❘}^{2},} & \left( {G5} \right)\end{matrix}$

where

${Q_{c} = \frac{R_{Q}}{16\pi R}}.$

An upper-bound for the resistance R can be found using the plasmon T₁ of10 μs, corresponding to a total quality factor of 1.86×10⁵, andQ_(c)=7.4×10⁴. This results in a fluxon T₁ limit in excess of 60 ms atthe flux-frustration point.

Relaxation from Dielectric Loss in the Capacitor

Dielectric loss associated with the capacitor can be thought of asJohnson-Nyquist current noise from the resistive part of the shuntingcapacitor, which couples to the phase matrix element

g|{circumflex over (φ)}|e

. This loss rate is therefore inversely proportional to the impedance ofthe capacitor, assuming a fixed loss tangent (1/Q_(diel)) for thecapacitor. As a result,

$\begin{matrix}{{{{S_{f}(\omega)} + {S_{f}\left( {- \omega} \right)}} = {\frac{{\hslash\omega}^{2}C}{Q_{diel}}\coth\left( \frac{\beta\hslash\omega}{2} \right)}},{{{and}\Gamma_{diel}} = {\frac{{\hslash\omega}^{2}}{8E_{C}Q_{cap}}{\coth\left( \frac{\beta\hslash\omega}{2} \right)}{{❘\left\langle {{\mathcal{g}0}{❘\hat{\phi}❘}{\mathcal{g}1}} \right\rangle ❘}^{2}.}}}} & ({G6})\end{matrix}$

If the T₁ at the frustration point were limited by dielectric loss, abath temperature of 42 mK would result in Q_(cap)=1/(8×10⁻⁶). This isclose to the expected loss tangent and within a factor of two of thatobserved in similar fluxonium devices. This is believed to be thedominant loss channel near the frustration point, also capturing theflux/frequency dependence of the measured loss (∝1/ω).

Relaxation from Dielectric Loss in the Inductor

For inductive loss, we again assume a frequency independent loss tangent(L→L(1+i/Q_(ind))), resulting in Johnson-Nyquist current noise that isinversely proportional to the impedance of the superinductor, i.e.,

${{S_{f}(\omega)} + {S_{f}\left( {- \omega} \right)}} = {\frac{\hslash}{{LQ}_{ind}}{{\coth\left( \frac{\beta\hslash\omega}{2} \right)}.}}$

the inductive loss is thus,

$\begin{matrix}{\Gamma_{ind} = {\frac{E_{L}}{\hslash Q_{L}}{\coth\left( \frac{\beta\hslash\omega}{2} \right)}{{❘\left\langle {{\mathcal{g}0}{❘\hat{\phi}❘}{\mathcal{g}1}} \right\rangle ❘}^{2}.}}} & ({G7})\end{matrix}$

The superinductor is extremely low loss, with a quality factor ofQ_(ind)=5×10⁹ resulting in a limit of T₁=2 ms at the flux frustrationpoint, growing as ω³ as we move away from the flux-frustration point.

Relaxation Rate Due to the Purcell Effect

We derive the Purcell relaxation rates of the fluxonium levels, arisingfrom coupling to the resonator. We model this by assuming that theresonator is coupled to a bath of harmonic oscillators, whoseHamiltonian reads

$\begin{matrix}{{H_{bath} = {\sum\limits_{k}{{\hslash\omega}_{k}b_{k}^{\dagger}b_{k}}}},} & ({G8})\end{matrix}$

where b_(k) is the lowering operator for mode k. The interactionHamiltonian between the bath and the resonator is given by

$\begin{matrix}{{H_{int} = {\hslash{\sum\limits_{k}{\lambda_{k}\left( {{ab}_{k}^{\dagger} + {a^{\dagger}b_{k}}} \right)}}}},} & ({G9})\end{matrix}$

where a is the lowering operator for the resonator. Finally, the systemunder consideration is the fluxonium circuit coupled to the resonator,which we write in the dressed basis as

$\begin{matrix}{{{\left. {H_{{flux} + {res}} = {\sum\limits_{k}{E_{k}^{{flux} + {res}}{❘\psi_{k}^{{flux} + {res}}}}}} \right\rangle\left\langle \psi_{k}^{{flux} + {res}} \right.}❘}.} & ({G10})\end{matrix}$

We treat H_(int) as a perturbation which can induce transitions amongthe eigenstates of the Hamiltonian H=H_(bath)+H_(flux+res), given by

$\begin{matrix}{\left. {\left. {\left. {❘\psi_{i}} \right\rangle = {❘\psi_{i}^{{flux} + {res}}}} \right\rangle\underset{k}{\otimes}{❘m_{k}}} \right\rangle.} & ({G11})\end{matrix}$

The transition rate under the action of a constant perturbation is givenby Fermi's Golden Rule in the form

$\begin{matrix}{{\gamma_{i\rightarrow f} = {\frac{2\pi}{\hslash}{\delta\left( {E_{i} - E_{f}} \right)}{❘\left\langle {\psi_{f}{❘H_{int}❘}\psi_{i}} \right\rangle ❘}^{2}}},} & ({G12})\end{matrix}$

where E_(i) and E_(f) are the eigenenergies of the states |ψ_(i)

and |ψ_(f)

, respectively. These energies are

$\begin{matrix}{{E_{i} = {E_{i}^{{flux} + {res}} + {\hslash{\sum\limits_{k}{m_{k}\omega_{k}}}}}},} & ({G13})\end{matrix}$${E_{f} = {E_{f}^{{flux} + {res}} + {\hslash{\sum\limits_{k}{m_{k}^{\prime}\omega_{k}}}}}},$

where {m_(k)} denotes the initial configuration of the bath and {m_(k)′}the final configuration. Inserting the form of H_(int) into Eqn. G12 andnoting that cross-terms vanish leads to

$\begin{matrix}{\gamma_{i,{{\{ m_{k}\}}\rightarrow f},{\{ m_{k^{\prime}}\}}} = {2{{\pi\hslash\delta}\left( {E_{i} - E_{f}} \right)}{\sum\limits_{k}{{❘\lambda_{k}❘}^{2}\left( {{{❘\left\langle {\psi_{f}^{{flux} + {res}}{❘a^{\dagger}❘}\psi_{i}^{{flux} + {res}}} \right\rangle ❘}^{2}m_{k}\delta_{m_{k^{\prime}},{m_{k} - 1}}} + {{❘\left\langle {\psi_{f}^{{flux} + {res}}{❘a❘}\psi_{i}^{{flux} + {res}}} \right\rangle ❘}^{2}\left( {m_{k} + 1} \right)\delta_{m_{k^{\prime}},{m_{k} + 1}}}} \right){\prod\limits_{k^{\prime} \neq k}\delta_{m_{k^{\prime}}}}}}}} & ({G14})\end{matrix}$

To find the total transition rate, we must sum over all such initial andfinal configurations, taking into account the thermal probability ofoccupying a given initial configuration:

$\begin{matrix}{{\Gamma_{i\rightarrow f} = {\sum\limits_{{\{ m_{k}\}},{\{ m_{k^{\prime}}\}}}{{P\left( \left\{ m_{k} \right\} \right)}\gamma_{i,{{\{ m_{k}\}}\rightarrow f},{\{ m_{k^{\prime}}\}}}}}},} & ({G15})\end{matrix}$ where $\begin{matrix}{{{P\left( \left\{ m_{k} \right\} \right)} = \frac{e^{- {\sum_{k}{\beta_{m_{k}}{\hslash\omega}_{k}}}}}{Z}},} & ({G16})\end{matrix}$

Z is the partition function of the bath and β=1/k_(B)T. Performing thesums over all initial and final states yields

$\begin{matrix}{{\Gamma_{i\rightarrow f} = {{2{\pi\hslash}{\sum\limits_{k}{{❘\lambda_{k}❘}^{2}{\delta\left( {E_{i}^{{flux} + {res}} - E_{f}^{{flux} + {res}} + {\hslash\omega}_{k}} \right)}{❘\left\langle {\psi_{f}^{{flux} + {res}}{❘a^{\dagger}❘}\psi_{i}^{{flux} + {res}}} \right\rangle ❘}^{2}{n_{th}\left( \omega_{k} \right)}}}} + {2{\pi\hslash}{\sum\limits_{k}{{❘\lambda_{k}❘}^{2}{\delta\left( {E_{i}^{{flux} + {res}} - E_{f}^{{flux} + {res}} - {\hslash\omega}_{k}} \right)}{❘\left\langle {\psi_{f}^{{flux} + {res}}{❘a❘}\psi_{i}^{{flux} + {res}}} \right\rangle ❘}^{2}\left( {{n_{th}\left( \omega_{k} \right)} + 1} \right)}}}}},} & ({G17})\end{matrix}$ where $\begin{matrix}{{n_{th}\left( \omega_{j} \right)} = {{\sum\limits_{\{ m_{k}\}}{{P\left( \left\{ m_{k} \right\} \right)}m_{j}}} = {\frac{1}{e^{{\beta\hslash\omega}_{j}} - 1}.}}} & ({G18})\end{matrix}$

We next take the continuum limit and define κ=2πℏρ(ω_(k))|λ_(k)|² whereρ(ω) is the density of states of the bath. Introducing ω_(jj′)^(flux+res)=(E_(j) ^(flux+res)−E_(j′) ^(flux+res))/ℏ leads to theexpressions

Γ_(i→f) ^(↑) =κn _(th)(ω_(fi) ^(flux+res))|

ψ_(f) ^(flux+res) |a ^(†)|ψ_(i) ^(flux+res)

|²,  (G19)

for upward transitions E_(f) ^(flux+res)>E_(i) ^(flux+res), and

Γ_(i→f) ^(↓)=κ(n _(th)(−ω_(fi) ^(flux+res))+1)|

ψ_(f) ^(flux+res) |a|ψ _(i) ^(flux+res)

|²,  (G20)

for downward transitions E_(f) ^(flux+res)≤E_(i) ^(flux+res). The finalstep is to note that throughout this experiment, the fluxonium qubit isoperated in the dispersive regime with respect to the frequency of theresonator. Therefore, we expect that the dressed eigenstates ofH_(flux+res) can be labeled with quantum numbers

and n, with

labeling the fluxonium state and n the resonator state. When performingnumerical simulations, this identification is based on which numbers

and n produce the maximum overlap of the dressed state |ψ_(i)^(flux+res)

=

with the product state |

,n

. We are interested mainly in transitions among fluxonium states, wherethe quantum number

changes. We therefore define the total transition rate due to thePurcell effect among fluxonium states as a sum over all possible initialand final states of the resonator, weighting initial states by theirprobability of being thermally occupiedP_(res)(n)=(1−exp(−βℏω_(r)))exp(−nβℏω_(r)). This yields

$\begin{matrix}{{\Gamma_{\ell\rightarrow\ell^{\prime}}^{{Purcell}, \uparrow} = {\sum\limits_{n,n^{\prime}}{{P_{res}(n)}\kappa{n_{th}\left( \omega_{\ell^{\prime},n^{\prime},\ell,n} \right)} \times {❘\left\langle {\overset{\_}{\ell^{\prime},n^{\prime}}{❘a^{\dagger}❘}\overset{\_}{\ell,n}} \right\rangle ❘}^{2}}}},} & ({G21})\end{matrix}$

for upward transitions, where

=(

−

)/ℏ, and

$\begin{matrix}{{\Gamma_{\ell\rightarrow\ell^{\prime}}^{{Purcell}, \downarrow} = {\sum\limits_{n,n^{\prime}}{{P_{res}(n)}\kappa\left( {{n_{th}\left( {- \omega_{\ell^{\prime},n^{\prime},\ell,n}} \right)} + 1} \right) \times {❘\left\langle {\overset{\_}{\ell^{\prime},n^{\prime}}{❘a❘}\overset{\_}{\ell,n}} \right\rangle ❘}^{2}}}},} & ({G22})\end{matrix}$

for downward transitions. The direct Purcell loss (|e

→|g

) gives a T₁ limit ˜100 ms, effectively negligible in our experiments.However, heating to the excited levels of fluxonium due to the finitebath temperature, results in enhanced Purcell loss. Some of these states(8^(th), 9^(th) and 10^(th) eigenstates) have transition frequenciesfrom the logical manifold that are close to the resonator frequency,resulting in avoided crossings. While their exact location dependssensitively on the circuit parameters, these resonances are likelyresponsible for the decreased T₁ observed near 0.35Φ₀. The total Purcellrelaxation rate for a bath temperature of 60 mK corresponds the dottedcurve in FIG. 9A.

Modeling Fluxonium Dephasing

On the flux slope, the decay envelope of a Ramsey experiment is bestapproximated by a gaussian exp(−t²/T_(ϕ) ²), where T_(ϕ)=Γ_(ϕ)⁻¹=(√{square root over (2)}η(∂_(ϕ)ω₀₁)√{square root over (lnω_(ir)t)})⁻¹ to first order. For the spin-echo experiments,low-frequency noise has a reduced weight in the noise spectrum, withT_(ϕ)=(√{square root over (W)}η(∂_(ϕ)ω₀₁))⁻¹. At the flux frustrationpoint, the qubit is first-order insensitive to 1/f flux noise, and thespin-echo data can be explained with an exponential decay from whitenoise (T_(2e)=T_(C)=Γ_(C) ⁻¹). In the regime of our spin-echo fluxsweep, both noise sources contribute significantly. The data istherefore fit to a product of a gaussian and an exponential, with theT_(2e) defined as exp(−T_(2e)/T_(C)−T_(2e) ²/T_(ϕ) ²)=1/e, i.e.,

$\begin{matrix}{T_{2e} = {\frac{\sqrt{{1/T_{C}^{2}} + {4/T_{\phi}^{2}}} - {1/T_{C}}}{2/T_{\phi}^{2}}.}} & ({H1})\end{matrix}$

CONCLUSION

We have realized a heavy-fluxonium qubit with a 14 MHz transitionfrequency and coherence times exceeding those of state-of-the-arttransmons, while demonstrating protocols for plasmon-assisted reset andreadout of the qubit, and a new flux control scheme that performs fasthigh-fidelity gates. We have explored a new frequency regime insuperconducting qubits and demonstrated the feasibility of a sub-thermalfrequency qubit, providing a path for manipulating fluxonium qubits withcomputational frequencies in the range of several GHz at temperaturesmuch higher than current dilution-refrigerator temperatures. Our newcontrol scheme has dramatically improved the single-qubit gate speed offluxonium qubits, making them a viable candidate for large-scalesuperconducting quantum computation. The gate pulses can be directlysynthesized with inexpensive digital to analog converters, and areinsensitive to shape distortions. Furthermore, the single-qubit gatescheme used in this work can be generalized to two inductively coupledfluxonium circuits, allowing for two-qubit gate operations withoutinvolving the participation of excited levels with more loss.

COMBINATION OF FEATURES

Features described above as well as those claimed below may be combinedin various ways without departing from the scope hereof. The followingexamples illustrate possible, non-limiting combinations of features andembodiments described above. It should be clear that other changes andmodifications may be made to the present embodiments without departingfrom the spirit and scope of this invention:

(A1) A method for initializing a quantum system formed from a qubitcoupled to an energy dissipater includes exciting the quantum systemfrom a first quantum state to a second quantum state. The qubit has aqubit ground state, a qubit metastable state, a first qubit excitedstate, and a second qubit excited state lying above the first qubitexcited state. The first quantum state is a composite of the qubitground state and a dissipater ground state of the energy dissipater, andthe second quantum state is a composite of the second qubit excitedstate and the dissipater ground state. The method also includes couplingthe quantum system from the second quantum state to a third quantumstate that is a composite of the qubit metastable state and a dissipaterexcited state of the energy dissipater. The quantum system decays fromthe third quantum state to a fourth quantum state that is a composite ofthe qubit metastable state and the dissipater ground state.

(A2) In the method denoted A1, the qubit may be a flux qubit.

(A3) In the method denoted A2, the flux qubit may be a fluxonium qubit.

(A4) In either one of the methods denoted A2 and A3, the flux qubit maybe a fluxonium qubit.

(A5) In any one of the methods denoted A2 to A4, the qubit ground stateand the qubit metastable state may be connected via a fluxon-liketransition, the qubit ground state and the second qubit excited statemay be connected via a plasmon-like transition, and the qubit metastablestate and the first qubit excited state may be connected via aplasmon-like transition.

(A6) In any one of the methods denoted A2 to A5, the method furtherincludes threading the flux qubit with magnetic flux to form the qubitground state, the qubit metastable state, the first qubit excited state,and the second qubit excited state

(A7) In the method denoted A6, the magnetic flux may be one-half of asuperconducting magnetic flux quantum.

(A8) In either one of the methods denoted A6 and A7, said threading,said exciting, and said coupling may occur simultaneously.

(A9) In the method denoted A8, a duration of said threading, saidexciting, and said coupling may be less than both a qubit relaxationtime and a qubit dephasing time of the flux qubit.

(A10) In any one of the methods denoted A1 to A9, the energy dissipatermay be a resonator or a transmission line.

(A11) In any one of the methods denoted A1 to A10, said exciting mayinclude driving the quantum system with a first microwave field having afirst frequency that is resonant with a first transition between thefirst and second quantum states. Said coupling may include driving thequantum system with a second microwave field having a second frequencythat is resonant with a second transition between the second and thirdquantum states.

(A12) In any one of the methods denoted A1 to A11, the method mayfurther include transferring, with a pi pulse, the quantum system fromthe fourth quantum state to the first quantum state.

(A13) In the method denoted A12, a frequency of the pi pulse may beresonant with a transition between the fourth and first quantum states.

(A14) In any one of the methods denoted A1 to A13, the method mayfurther include preparing, prior to said exciting and said coupling, thequantum system such that the qubit is in a thermal mixed state of thequbit ground state and the qubit metastable state. The quantum system,after subsequently decaying from the third quantum state to the fourthquantum state, may then be in an approximately pure quantum state.

(A15) In any one of the methods denoted A1 to A14, a qubit transitionfrequency between the qubit ground state and the qubit metastable statemay be less than or equal to a temperature of a thermal bath surroundingthe quantum system.

(A16) In any one of the methods denoted A1 to A15, the qubit metastablestate may have a significant thermal occupation prior to said exciting.

(A17) In any one of the methods denoted A1 to A16, the method mayfurther include cryogenically cooling the quantum system such that atemperature of a thermal bath surrounding the quantum system is greaterthan or equal to a qubit transition frequency between the qubit groundstate and the qubit metastable state.

(B1) A method for initializing a quantum system formed from a qubitcoupled to an energy dissipater includes exciting the quantum systemfrom a first quantum state to a second quantum state. A qubit transitionfrequency between a qubit ground state of the qubit and a qubitmetastable state of the qubit is less than or equal to a temperature ofa thermal bath surrounding the quantum system. The first quantum stateis a composite of the qubit ground state and a dissipater ground stateof the energy dissipater, and the second quantum state is a composite ofa second qubit excited state and the dissipater ground state. The methodalso includes coupling the quantum system from the second quantum stateto a third quantum state that is a composite of the qubit metastablestate and a dissipater excited state of the energy dissipater.

(B2) In the method denoted B1, the qubit may be a flux qubit.

(B3) In the method denoted B2, the flux qubit may be a fluxonium qubit.

(B4) In either one of the methods denoted B2 and B3, the flux qubit maybe a heavy fluxonium qubit.

(B5) In any one of the methods denoted B2 to B4, the qubit ground stateand the qubit metastable state may be connected via a fluxon-liketransition. The qubit ground state and the qubit excited state may beconnected via a plasmon-like transition.

(B6) In any one of the methods denoted B2 to B5, the method may furtherinclude threading the flux qubit with magnetic flux to form the qubitground state, the qubit metastable state, and the qubit excited state.

(B7) In the method denoted B6, the magnetic flux may be one-half of asuperconducting magnetic flux quantum.

(B8) In either one of the methods denoted B6 and B7, said threading,said exciting, and said coupling may occur simultaneously.

(B9) In the method denoted B8, wherein a duration of said threading,said exciting, and said coupling may be less than both a qubitrelaxation time and a qubit dephasing time of the flux qubit.

(B10) In any one of the methods denoted B1 to B9, the energy dissipatermay be a resonator or a transmission line.

(B11) In any one of the methods denoted B1 to B10, said exciting mayinclude driving the quantum system with a first microwave field having afirst frequency that is resonant with a first transition between thefirst and second quantum states. Said coupling may include driving thequantum system with a second microwave field having a second frequencythat is resonant with a second transition between the second and thirdquantum states.

(B12) In any one of the methods denoted B1 to B11, the quantum systemdecays from the third quantum state to a fourth quantum state that is acomposite of the qubit metastable state and the dissipater ground state.The method may further include transferring, with a pi pulse, thequantum system from the fourth quantum state to the first quantum state.

(B13) In the method denoted B12, a frequency of the pi pulse may beresonant with a transition between the fourth and first quantum states.

(B14) In any one of the methods denoted B1 to B13, the method mayfurther include preparing, prior to said exciting and said coupling, thequantum system such that the qubit is in a thermal mixed state of thequbit ground state and the qubit metastable state. The quantum system,after subsequently decaying from the third quantum state to a fourthquantum state that is a composite of the qubit metastable state and thedissipater ground state, is in an approximately pure quantum state.

(B15) In any one of the methods denoted B1 to B14, the qubit metastablestate may have a significant thermal occupation prior to said exciting.

(B16) In any one of the methods denoted B1 to B15, the method mayfurther include cryogenically cooling the quantum system such that thetemperature of the thermal bath surrounding the quantum system isgreater than or equal to the qubit transition frequency.

(C1) A method for measuring a qubit state of a quantum system comprisinga qubit coupled to a resonator includes coupling a first quantum stateof the quantum system to a second quantum state of the quantum system.The qubit has a qubit ground state, a qubit metastable state, a firstqubit excited state, and a second qubit excited state lying above thefirst qubit excited state. The qubit state is a linear superposition ofthe qubit ground state and the qubit metastable state, the first quantumstate is a composite of the qubit metastable state and a resonatorground state of the resonator, and the second quantum state is acomposite of the first qubit excited state and the resonator groundstate. The method also includes dispersively reading the qubit statewith the resonator. A resonant frequency of the resonator is greaterthan a transition frequency between the qubit metastable state and thefirst qubit excited state.

(C2) In the method denoted C1, the qubit may be a flux qubit.

(C3) In the method denoted C2, the flux qubit may be a fluxonium qubit.

(C4) In either one of the methods denoted C2 and C3, the flux qubit maybe a heavy fluxonium qubit.

(C5) In any one of the methods denoted C2 to C4, the qubit ground stateand the qubit metastable state may be connected via a fluxon-liketransition, the qubit ground state and the second qubit excited statemay be connected via a plasmon-like transition, and the qubit metastablestate and the first qubit excited state may be connected via aplasmon-like transition.

(C6) In any one of the methods denoted C2 to C5, the method may furtherinclude threading the flux qubit with magnetic flux to form the qubitground state, the qubit metastable state, the first qubit excited state,and the second qubit excited state.

(C7) In the method denoted C6, the magnetic flux may be one-half of asuperconducting magnetic flux quantum.

(C8) In any one of the methods denoted C2 to C7, a qubit transitionfrequency between the qubit ground state and the qubit metastable statemay be less than a temperature of a thermal bath surrounding the quantumsystem.

(C9) In any one of the methods denoted C2 to C8, said coupling mayinclude applying a pi pulse.

(C10) In the method denoted C9, a frequency of the pi pulse may beresonant with a transition between the first and second quantum states.

(C11) In any one of the methods denoted C2 to C10, the method mayfurther include preparing, prior to said coupling and said dispersivelyreading, the quantum system such that the qubit is in the qubit state.

(D1) A method for measuring a qubit state of a quantum system comprisinga qubit coupled to a resonator includes coupling a first quantum stateof the quantum system to a second quantum state of the quantum system.The qubit has a qubit ground state, a qubit metastable state, a firstqubit excited state, and a second qubit excited state lying above thefirst qubit excited state. The qubit state is a linear superposition ofthe qubit ground state and the qubit metastable state, the first quantumstate is a composite of the qubit ground state and a resonator groundstate of the resonator, and the second quantum state is a composite ofthe second qubit excited state and the resonator ground state. Themethod also includes dispersively reading the qubit state with theresonator. A resonant frequency of the resonator is greater than atransition frequency between the qubit ground state and the second qubitexcited state.

(D2) In the method denoted D1, the qubit may be a flux qubit.

(D3) In the method denoted D2, the flux qubit may be a fluxonium qubit.

(D4) In either one of the methods denoted D2 and D3, the flux qubit maybe a heavy fluxonium qubit.

(D5) In any one of the methods denoted D2 to D4, the qubit ground stateand the qubit metastable state may be connected via a fluxon-liketransition, the qubit ground state and the second qubit excited statemay be connected via a plasmon-like transition, and the qubit metastablestate and the first qubit excited state may be connected via aplasmon-like transition.

(D6) In any one of the methods denoted D2 to D5, the method may furtherinclude threading the flux qubit with magnetic flux to form the qubitground state, the qubit metastable state, the first qubit excited state,and the second qubit excited state.

(D7) In the method denoted D6, the magnetic flux may be one-half of asuperconducting magnetic flux quantum.

(D8) In any one of the methods denoted D1 to D7, a qubit transitionfrequency between the qubit ground state and the qubit metastable statemay be less than a temperature of a thermal bath surrounding the quantumsystem.

(D9) In any one of the methods denoted D1 to D8, said coupling mayinclude applying a pi pulse.

(D10) In the method denoted D9, a frequency of the pi pulse may beresonant with a transition between the first and second quantum states.

(D11) In any one of the methods denoted D1 to D10, the method mayfurther include preparing, prior to said coupling and said dispersivelyreading, the quantum system such that the qubit is in the qubit state.

(E1) A method for rotating a flux qubit in a linear superposition offirst and second quantum-computational states includes threading theflux qubit with a magnetic flux at a nominal value such that the firstand second quantum-computational states are separated by a nominalenergy splitting. The linear superposition may be represented as a Blochvector on a Bloch sphere. The method includes applying a first pulse tothe flux qubit by momentarily deviating the magnetic flux away from thenominal value to rotate the Bloch vector about an x axis of the Blochsphere.

(E2) In the method denoted E1, the first pulse may rotate the Blochvector about the x axis by a first angle, and about a z axis of theBloch sphere.

(E3) In either one of the methods denoted E1 and E2, a duration of thefirst pulse may be less than two periods of the energy splitting.

(E4) In any one of the methods denoted E1 to E3, the first and secondquantum-computational states may be separated by a maximum energysplitting occurring at a peak of the first pulse, and an amplitude ofthe first pulse may be selected such that the maximum energy splittingis at least twice the nominal energy splitting.

(E5) In any one of the methods denoted E1 to E4, the nominal value ofthe magnetic flux may be one-half of a superconducting magnetic fluxquantum, wherein the nominal energy splitting is equal to a Larmorfrequency of the flux qubit.

(E6) In any one of the methods denoted E1 to E5, said applying the firstpulse may rotate the Bloch vector about the x axis of the Bloch sphereby a first angle. The method may further include idling, after saidapplying the first pulse, with the magnetic flux at the nominal value torotate the Bloch vector by a second angle about a z axis of the Blochsphere. The method may also include applying, after said idling, asecond pulse that rotates the Bloch vector by the negative of the firstangle about the x axis of the Bloch sphere.

(E7) In the method denoted E6, an area of the first pulse and an area ofsecond pulse may add to zero.

(E8) In either one of the methods denoted E6 and E7, a duration of thefirst pulse may equal a duration of the second pulse.

(E9) In any one of the methods denoted E6 to E8, the first and secondpulses may deviate the magnetic flux away from the nominal value inopposite directions.

(E10) In any one of the methods denoted E6 to E9, a duration of each ofthe first and second pulses may be shorter than a duration of saididling.

(E11) In any one of the methods denoted E6 to E10, said applying thefirst pulse may additionally rotate the Bloch vector by the first angleabout the z axis of the Bloch sphere. Said applying the second pulse mayadditionally rotate the Bloch vector by the first angle about the z axisof the Bloch sphere.

(E12) In any one of the methods denoted E6 to E11, the first pulse, thesecond pulse, and a duration of said idling may be configured such thatthe Bloch vector rotates about a y axis of the Bloch sphere by eitherpositive ninety degrees or negative ninety degrees.

(E13) In any one of the methods denoted E1 to E12, the flux qubit may bea fluxonium qubit.

(E14) In any one of the methods denoted E1 to E13, the flux qubit may bea heavy fluxonium qubit.

(F1) A method for rotating a flux qubit by an angle includes threadingthe flux qubit with a magnetic flux at a nominal value such that firstand second quantum-computational states of the flux qubit are separatedby a nominal energy splitting. The flux qubit is in a superposition ofthe first and second quantum-computational states, and the linearsuperposition is represented as a Bloch vector on a Bloch sphere. Themethod also includes applying a first pulse to the flux qubit bymomentarily deviating the magnetic flux away from the nominal value torotate the Bloch vector by negative ninety degrees about a y axis of theBloch sphere. The method also includes idling, after said applying thefirst pulse, the magnetic flux at the nominal value to rotate the Blochvector by the angle about a z axis of the Bloch sphere. The method alsoincludes applying, after said idling, a second pulse to the flux qubitto rotate the Bloch vector by positive ninety degrees about the y axisof the Bloch sphere. The Bloch vector is rotated by the angle about an xaxis of the Bloch sphere.

(F2) In the method denoted F1, the flux qubit may be a fluxonium qubit.

(F3) In either one of the methods denoted F1, the flux qubit may be aheavy fluxonium qubit.

Changes may be made in the above methods and systems without departingfrom the scope hereof. It should thus be noted that the matter containedin the above description or shown in the accompanying drawings should beinterpreted as illustrative and not in a limiting sense. The followingclaims are intended to cover all generic and specific features describedherein, as well as all statements of the scope of the present method andsystem, which, as a matter of language, might be said to falltherebetween.

1. A method for initializing a quantum system formed from a qubitcoupled to an energy dissipater, the qubit having a qubit ground state,a qubit metastable state, a first qubit excited state, and a secondqubit excited state lying above the first qubit excited state, saidmethod comprising: exciting the quantum system from a first quantumstate to a second quantum state, the first quantum state being acomposite of the qubit ground state and a dissipater ground state of theenergy dissipater, the second quantum state being a composite of thesecond qubit excited state and the dissipater ground state; and couplingthe quantum system from the second quantum state to a third quantumstate that is a composite of the qubit metastable state and a dissipaterexcited state of the energy dissipater; wherein the quantum systemdecays from the third quantum state to a fourth quantum state that is acomposite of the qubit metastable state and the dissipater ground state.2. The method of claim 1, the qubit being a flux qubit.
 3. The method ofclaim 2, the flux qubit being a fluxonium qubit.
 4. The method of claim2, the flux qubit being a heavy fluxonium qubit.
 5. The method of claim2, wherein: the qubit ground state and the qubit metastable state areconnected via a fluxon-like transition; the qubit ground state and thesecond qubit excited state are connected via a plasmon-like transition;and the qubit metastable state and the first qubit excited state areconnected via a plasmon-like transition.
 6. The method of claim 2,further comprising threading the flux qubit with magnetic flux to formthe qubit ground state, the qubit metastable state, the first qubitexcited state, and the second qubit excited state.
 7. The method ofclaim 6, the magnetic flux being one-half of a superconducting magneticflux quantum.
 8. The method of claim 6, wherein said threading, saidexciting, and said coupling occur simultaneously.
 9. The method of claim8, wherein a duration of said threading, said exciting, and saidcoupling is less than both a qubit relaxation time and a qubit dephasingtime of the flux qubit.
 10. The method of claim 1, the energy dissipaterbeing a resonator or a transmission line.
 11. The method of claim 1,wherein: said exciting includes driving the quantum system with a firstmicrowave field having a first frequency that is resonant with a firsttransition between the first and second quantum states; and saidcoupling includes driving the quantum system with a second microwavefield having a second frequency that is resonant with a secondtransition between the second and third quantum states.
 12. The methodof claim 1, further comprising transferring, with a pi pulse, thequantum system from the fourth quantum state to the first quantum state.13. The method of claim 12, wherein a frequency of the pi pulse isresonant with a transition between the fourth and first quantum states.14. The method of claim 1, further comprising preparing, prior to saidexciting and said coupling, the quantum system such that the qubit is ina thermal mixed state of the qubit ground state and the qubit metastablestate; wherein the quantum system, after subsequently decaying from thethird quantum state to the fourth quantum state, is in an approximatelypure quantum state.
 15. The method of claim 1, wherein a qubittransition frequency between the qubit ground state and the qubitmetastable state is less than or equal to a temperature of a thermalbath surrounding the quantum system.
 16. The method of claim 1, whereinthe qubit metastable state has a significant thermal occupation prior tosaid exciting.
 17. The method of claim 1, further comprisingcryogenically cooling the quantum system such that a temperature of athermal bath surrounding the quantum system is greater than or equal toa qubit transition frequency between the qubit ground state and thequbit metastable state.
 18. A method for initializing a quantum systemformed from a qubit coupled to an energy dissipater, wherein a qubittransition frequency between a qubit ground state of the qubit and aqubit metastable state of the qubit is less than or equal to atemperature of a thermal bath surrounding the quantum system, saidmethod comprising: exciting the quantum system from a first quantumstate to a second quantum state, the first quantum state being acomposite of the qubit ground state and a dissipater ground state of theenergy dissipater, the second quantum state being a composite of a qubitexcited state of the qubit and the dissipater ground state; and couplingthe quantum system from the second quantum state to a third quantumstate that is a composite of the qubit metastable state and a dissipaterexcited state of the energy dissipater.
 19. The method of claim 18, thequbit being a flux qubit. 20-22. (canceled)
 23. The method of claim 19,further comprising threading the flux qubit with magnetic flux to formthe qubit ground state, the qubit metastable state, and the qubitexcited state. 24-72. (canceled)